Second cohomology group for trivial group action of E8 on V4

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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 elementary abelian group:E8 on Klein four-group. The elements of this classify the group extensions with Klein four-group in the center and elementary abelian group:E8 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:E4096.
Get more specific information about elementary abelian group:E8 |Get more specific information about Klein four-group|View other constructions whose value is elementary abelian group:E4096

Description of the group

This group is defined as the second cohomology group for trivial group action of elementary abelian group:E8 on Klein four-group, i.e., the group:

\! H^2(G,A)

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

The group is an elementary abelian group of order 2^{12} = 4096.

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 Yes
symmetric nontrivial direct product of E8 and Z4 45 No
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 SmallGroup(32,24) 24 Yes
non-symmetric direct product of D8 and Z4 25 Yes
non-symmetric direct product of Q8 and Z4 26 Yes
non-symmetric SmallGroup(32,27) 27 Yes
non-symmetric SmallGroup(32,28) 28 Yes
non-symmetric SmallGroup(32,29) 29 Yes
non-symmetric SmallGroup(32,30) 30 Yes
non-symmetric SmallGroup(32,31) 31 Yes
non-symmetric SmallGroup(32,32) 32 Yes
non-symmetric SmallGroup(32,33) 33 Yes
non-symmetric SmallGroup(32,34) 34 Yes
non-symmetric SmallGroup(32,35) 35 Yes
non-symmetric direct product of D8 and V4 46 No
non-symmetric direct product of Q8 and V4 47 No
non-symmetric direct product of SmallGroup(16,13) and Z2 48 Yes