Second cohomology group for trivial group action of V4 on E8

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