Group cohomology of quaternion group

Over the integers
The homology groups are given as follows:

$$H_q(Q_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl}\mathbb{Z}, & q = 0 \\ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, & q \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & q \equiv 3 \pmod 4 \\ 0, & q \ne 0, q \mbox{ even} \\\end{array}\right.$$

The group is a finite group with periodic cohomology, in keeping with the other definition of being a group with periodic cohomology: every abelian subgroup is cyclic.

The first few homology groups are given below:

Over an abelian group
The first few homology groups with coefficients in an abelian group $$M$$ are given below:

Over an abelian group
The first few cohomology groups with coefficients in an abelian group $$M$$ are as follows:

Schur multiplier
The Schur multiplier, defined as second cohomology group for trivial group action $$H^2(G,\mathbb{C}^\ast)$$, and also as the second homology group $$H_2(G;\mathbb{Z})$$, is the trivial group.

Schur covering groups
Since the Schur multiplier is a trivial group, the Schur covering group of the quaternion group is the quaternion group itself.