Second cohomology group for trivial group action of E8 on V4

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