In looking at the book closely, I have found that mention of "The Council of the Home" comes when Equality 7-2521 speaks of the Home of the Street Sweepers. He says:
So we went to the Home of the Street Sweepers. It is a grey house on a narrow street. There is a sundial in its courtyard, by which the Council of the Home can tell the hours of the day and when to ring the bell.
This tells us that there are people who are in charge of running the Homes that the citizens of different vocations live in. There are different Homes mentioned throughout the first few chapters and each "Home" has a capital "H," which lends importance to the home. For instance, old people go to live in the Home of the Useless once they have turned forty. Actors live in the Home of the Actors. Each Home has a Council that runs its daily affairs. They are likely in charge of headcounts at night as well as other controlling factors against the citizens.