Question

What are the tasks/jobs each caste in the caste system carries out in Brave New World?

Accepted Answer

In Aldous Huxley's Brave New World, there are five genetically-dictated castes that inhabit the New World.  In the supposed utopia, these predestined humans are named after the letters of the Greek alphabet:  Alphas, Betas, Gammas, Deltas, and Epsilons. 
ALPHAS
The upper castes such as the Alpha Pluses and Alphas are bred for managerial positions.  They have been given creative powers, but they have been conditioned to abhor books and nature.  The Director of the Hatchery is an Alpha, for example, as are Bernard Marx and Helmholtz.
BETAS
Not as intelligent as Alphas, Betas also can be minuses and pluses.  They have had some decanting (reducing of the oxygen), but are able to hold jobs that require intelligence.  Lenina is a Beta.  Like other Betas and Alphas, Lenina is taught to hate Gammas, Epsilons, and Deltas.  She makes the comment that she is glad that she is a Beta

because they're so frightfully clever.  I'm really awfully glad I'm a Beta, because I don't work so hard.  And then we are much better than the Gammas and Deltas.  Gammas are stupid.  They all wear green, and Delta children wear khaki.  Oh no, I don't want to play with Delta children.   And Epsilons are still worse.  They're too stupid to be able..."

GAMMAS - this caste is dressed in green
DELTAS - this caste is dressed in khaki and other castes are conditioned to hate khaki.
EPSILONS - this group has very little intelligence.
The Gammas, the Deltas, and the Epsilons have the menial jobs.  There are many of them bred to do these jobs such as running elevators and cleaning floors.
All the groups are conditioned through electric shock and hypnopoedia, sleep conditioning.  Sexual entertainment is also encouraged in the new world.

Answer

The classes in Brave New World are named after the first five letters of the Greek alphabet: Alpha, Beta, Gamma, Delta, and Epsilon. Within these five categories, we hear pluses and minuses discussed, and even Alpha Double-Pluses. Their jobs are as follows.
Alpha
The Alpha caste consists of the leaders and the thinkers. They do all the intellectual work and occupy all the highest-level positions. At a lower level (Alpha-Minus), they are administrators and managers. The Warden of the Reservation is an Alpha-Minus.
Beta
The Beta caste performs lower-level managerial and higher-level technical functions. Many of them are women. Mechanics, laboratory technicians, and other skilled workers are drawn from the Betas.
Gamma
Gammas are top-level servants and semi-skilled workers. The Controller's butler, for instance, is a Gamma. While Alphas and Betas do essentially middle-class work, the Gammas are at the top of the working class.
Delta
The Deltas are perhaps the least differentiated of the castes. They tend to be grouped together with the Epsilons, and sometimes the Gammas as well, to mean "the bottom part of society, who cannot be expected to understand anything." Many of them seem to work on the production lines in factories, making basic components for machines.
Epsilon
We are told that the Epsilon mind is mature at ten years old and that they are the only illiterate class. The Epsilons perform the most basic tasks. Sewage workers are mentioned as being Epsilons; Epsilons also work as lower-level servants, such as porters.

Answer

The Alphas are the thinking class and seem to be made up entirely (or almost entirely) of men. These are the university-educated types who run society. They are the tallest group—or are supposed to be—reflecting their position of authority in society. As the text notes:

Gammas, Deltas and Epsilons had been to some extent conditioned to associate corporeal mass with social superiority.

The Betas, who seem to be primarily women, are the next highest caste. They have critical thinking skills to an extent but are not of the intellectual caliber of the alphas. They are the lab technicians, simple problem solvers, and semi-professional workers of the society, and they provide a convenient pool of sexually willing women for the Alphas to sleep with.
The Gammas are the highest level of the three lower (or mass) classes and are the highest caste to be birthed in identical batches. They are factory workers and, because of the amount of time they have to spend in an industrial setting, are conditioned to hate nature, as are the other lower castes.
Deltas are also factory workers, but they have lower intelligence than the Gammas. One group of Deltas is described as "almost noseless [and] black."
The Epsilons are the lowest and least intelligent caste in the World State. They are the shortest and darkest, too, reflecting the racism of Huxley's time period. They perform the most menial tasks in society and are the group that are "hatched" in the largest batches, as their individuation is unimportant.
The book does little to differentiate between the Gammas, Deltas, and Epsilons except to make them progressively smaller, darker, and less intelligent. They are all factory workers and menial laborers.

Brave New World

What are the tasks/jobs each caste in the caste system carries out in Brave New World?

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