Second cohomology group for trivial group action of E8 on Z2

Description of the group
We consider here the second cohomology group for trivial group action of elementary abelian group:E8 on the cyclic group:Z2, i.e.,

$$\! H^2(G,A)$$

where $$G \cong E_8 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_2$$.

The cohomology group is isomorphic to elementary abelian group:E64.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Here, the trace of a cohomology class is defined as the trace of any normalized 2-cocycle representing that cohomology class, written as a $$|G| \times |G|$$ matrix. This is well defined because all the normalized 2-coboundaries are IIP, i.e., they have zero trace.

This trace can also be interpreted as the number of cosets of $$A$$ in the big extension group that have elements of order $$4$$.

Note that under the group operation of $$H^2(G,A)$$, the trace map gets added up to parity. In other words, for cohomology classes $$c_1$$ and $$c_2$$, the trace of $$c_1 + c_2$$ has the same parity as the sum of the traces of $$c_1$$ and $$c_2$$.

Under the action of the automorphism group of the acting group
The acting group $$G$$, which is isomorphic to elementary abelian group:E8, has a huge automorphism group, general linear group:GL(3,2) of order $$168$$. Each cohomology class type in the table above is one orbit. In particular, the trivial group extension is the only fixed point, there are two orbits of size $$7$$, and there is one orbit each of size $$21$$ and $$28$$.

General background
We know from the general theory that there is a natural short exact sequence:

$$0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0$$

where the image of $$\operatorname{Ext}^1$$ is $$H^2_{sym}(G;A)$$, i.e., the group of cohomology classes represented by symmetric 2-cocycles. We also know, again from the general theory, that the short exact sequence above splits, i.e., $$H^2_{sym}(G;A)$$ has a complement inside $$H^2$$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case
For this choice of $$G$$ and $$A$$, the subgroup $$H^2_{CP}(G;A)$$ of cyclicity-preserving cohomology classes is trivial, hence the sum $$H^2_{sym}(G;A) + H^2_{CP}(G;A)$$ is $$H^2_{sym}(G;A)$$ and is not the whole group $$H^2(G;A)$$. Thus, $$H^2_{CP}(G;A)$$ is not the desired complement. For this choice of $$G$$ and $$A$$, the subgroup $$H^2_{CP}(G,A)$$ of cyclicity-preserving cohomology classes is trivial, hence the sum $$H^2_{sym}(G,A) + H^2_{CP}(G,A)$$ is $$H^2_{sym}(G,A)$$ and is not the whole group $$H^2(G,A)$$. (The same holds true if we replace $$H^2_{CP}(G,A)$$ with $$H^2_{IIP}(G,A)$$).

In fact, there is no subgroup complement to $$H^2_{sym}(G,A)$$ in $$H^2(G,A)$$ that is invariant under the action of $$\operatorname{Aut}(G) \times \operatorname{Aut}(A)$$.

Construction of the cohomology group and extensions
The cohomology group and the corresponding extensions can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF.

gap> G := ElementaryAbelianGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ , ,  ] ), collector := rec( relators := [ [ 0 ], [ [ 2, 1 ], 0 ], [ [ 3, 1 ], [ 3, 1 ], 0 ] ],     orders := [ 2, 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1 ] ], estack := [ ], pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 0, 0, 0 ], module := [ , ,  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6 ] ), cohom :=  -> ( GF(2)^6 )>, presentation := rec( group := , relators := [ f1^2, f1^-1*f2*f1*f2^-1, f2^2, f1^-1*f3*f1*f3^-1, f2^-1*f3*f2*f3^-1, f3^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions.

gap> G := ElementaryAbelianGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 16, 14 ], [ 16, 10 ], [ 16, 11 ], [ 16, 11 ], [ 16, 11 ], [ 16, 11 ], [ 16, 11 ], [ 16, 11 ], [ 16, 10 ], [ 16, 10 ], [ 16, 11 ], [ 16, 12 ],  [ 16, 13 ], [ 16, 13 ], [ 16, 13 ], [ 16, 13 ], [ 16, 11 ], [ 16, 13 ],  [ 16, 11 ], [ 16, 13 ], [ 16, 11 ], [ 16, 13 ], [ 16, 13 ], [ 16, 11 ],  [ 16, 11 ], [ 16, 13 ], [ 16, 11 ], [ 16, 13 ], [ 16, 13 ], [ 16, 11 ],  [ 16, 11 ], [ 16, 13 ], [ 16, 10 ], [ 16, 10 ], [ 16, 13 ], [ 16, 13 ],  [ 16, 11 ], [ 16, 12 ], [ 16, 13 ], [ 16, 13 ], [ 16, 10 ], [ 16, 10 ],  [ 16, 13 ], [ 16, 13 ], [ 16, 13 ], [ 16, 13 ], [ 16, 11 ], [ 16, 12 ],  [ 16, 11 ], [ 16, 13 ], [ 16, 13 ], [ 16, 11 ], [ 16, 11 ], [ 16, 13 ],  [ 16, 11 ], [ 16, 13 ], [ 16, 12 ], [ 16, 13 ], [ 16, 13 ], [ 16, 12 ],  [ 16, 13 ], [ 16, 12 ], [ 16, 13 ], [ 16, 12 ] ]

Under the action of the various automorphism groups
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := ElementaryAbelianGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 16, 14 ], [ 16, 10 ], [ 16, 11 ], [ 16, 12 ], [ 16, 13 ] ]