Second cohomology group for trivial group action of V4 on E8

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This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group Klein four-group on elementary abelian group:E8. The elements of this classify the group extensions with elementary abelian group:E8 in the center and Klein four-group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E512.
Get more specific information about Klein four-group |Get more specific information about elementary abelian group:E8|View other constructions whose value is elementary abelian group:E512

Description of the group

This is the second cohomology group for trivial group action where the acting group is Klein four-group and the group being acted upon is elementary abelian group:E8. In other words, it is the group:

\! H^2(G,A)

where G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 and A \cong E_8 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2.

The group is isomorphic to elementary abelian group:E512, i.e., it is an elementary abelian group of order 2^9 = 512.

Elements

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 32) Base characteristic in whole group?
trivial 1 elementary abelian group:E32 51 No
symmetric nontrivial direct product of Z4 and Z4 and Z2 21 No
symmetric nontrivial direct product of E8 and Z4 45 No
non-symmetric SmallGroup(32,2) 2 Yes
non-symmetric direct product of SmallGroup(16,3) and Z2 22 Yes
non-symmetric direct product of SmallGroup(16,4) and Z2 23 Yes
non-symmetric direct product of D8 and V4 46 Yes
non-symmetric direct product of Q8 and V4 47 Yes