Second cohomology group for trivial group action of E8 on V4

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 $$2^{12} = 4096$$.