According to The Communist Manifesto, would a communist society consist of classes?
According to Marx and Engels, a communist society would have no classes. The class struggles that typified all of previous history would end with the triumph of the proletariat. After that, there would be no need for classes because everyone would be equal and no one would be oppressed.
In The Communist Manifesto, Marx and Engels state this point quite clearly. They tell us that (I have added the bold face)
When, in the course of development, class distinctions have disappeared, and all production has been concentrated in the hands of a vast association of the whole nation, the public power will lose its political character.
To them, political power is the organized power of one class oppressing another class. During our time, the proletariat are oppressed by the bourgeoisie and we have class struggle. However, once the proletariat wins this struggle, they will not oppress anyoe and so there would be no politics and no classes. As the authors tell us, by the time it is over, the revolution will
From this, it is quite clear that Marx and Engels believe that a communist society will be a classless society.