Group cohomology of groups of order 27

With the exception of the zeroth homology group and cohomology group, all homology and cohomology groups over all possible abelian groups are 3-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.

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 $$G$$.