Group cohomology of symmetric group:S6

Classifying space and corresponding chain complex
The homology and cohomology groups are the same as the respective homology and cohomology groups of the configuration space of six unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real vector space on the Topology Wiki.

Computation of homology groups
The homology groups for trivial group action on the integers can be computed using the Hap package (if the package is installed but not automatically loaded, load it using LoadPackage("hap");), specifically its GroupHomology function. The function returns a list of numbers which are the orders of cyclic groups whose external direct product is the desired homology group.

First homology group
The first homology group, which is also the abelianization, can be computed as follows:

gap> GroupHomology(SymmetricGroup(6),1); [ 2 ]

This says that $$H_1(S_6;\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$$.

Second homology group
The second homology group, which is also the Schur multiplier, can be computed as follows:

gap> GroupHomology(SymmetricGroup(6),2); [ 2 ]

This says that $$H_2(S_6;\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}$$.

First few homology groups
gap> List([1..8],i -> [i,GroupHomology(SymmetricGroup(6),i)]); [ [ 1, [ 2 ] ], [ 2, [ 2 ] ], [ 3, [ 2, 2, 4, 3 ] ], [ 4, [ 2, 2 ] ], [ 5, [ 2, 2, 2, 2, 2 ] ], [ 6, [ 2, 2, 2, 2, 2 ] ],  [ 7, [ 2, 2, 2, 2, 2, 2, 4, 3, 3, 5 ] ], [ 8, [ 2, 2, 2, 2, 2, 2, 2, 2 ] ] ]