Group cohomology of groups of order 8

With the exception of the zeroth homology group and cohomology group, all homology and cohomology groups over all possible abelian groups are 2-groups.

Over the integers
The table below lists the first few homology groups with coefficients in the integers. We use $$\mathbb{Z}_n$$ to denote the cyclic group of order $$n$$.

We use 0 to denote the trivial group.

Over an abelian group
Below are the homology groups for trivial group action with coefficients in an abelian group $$M$$. We denote by $$aM$$ the subgroup $$\{ am \mid m \in M \}$$ and by $$\operatorname{Ann}_M(a)$$ the subgroup $$\{ x \in M \mid ax = 0 \}$$.

These groups can be computed from the homology groups with coefficients in the integers by using the universal coefficients theorem for group homology.

Over the integers
The table below lists the first few cohomology groups with coefficients in the integers. We use $$\mathbb{Z}_n$$ to denote the cyclic group of order $$n$$. Note that each cohomology group is just the previous homology group, i.e., $$H^q \cong H_{q-1}$$. This is a consequence of the dual universal coefficients theorem for group cohomology.

All the $$H^1$$ groups are trivial groups. We use 0 to denote the trivial group.

Schur multiplier and Schur covering groups
The Schur multiplier is defined as second cohomology group for trivial group action, $$H^2(G;\mathbb{C}^\ast)$$, and also as the second homology group $$H_2(G;\mathbb{Z})$$. A corresponding Schur covering group of $$G$$ is a group that arises as a stem extension with base normal subgroup the Schur multiplier and the quotient group is $$G$$.

Higher nilpotent multipliers
Note that nilpotent multiplier of nilpotent group for class higher than its class is trivial, so the 3-nilpotent multiplier and higher nilpotent multipliers are all trivial.