Group cohomology of groups of prime-cube order

This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of prime-cube order.
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This article describes the group cohomology of groups of prime-cube order. The two cases $p=2$ (see group cohomology of groups of order 8) and $p=3$ (see group cohomology of groups of order 27) are somewhat special.

Second cohomology groups and extensions

Schur multipliers 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 $G$.

Group	GAP ID 2nd part	Nilpotency class	$H_{2}$ (= Schur multiplier)	Order of $H_{2}$	Possibilities for Schur covering groups	Cohomology group information	Orders of Schur covering groups
cyclic group of prime-cube order	1	1	trivial group	1	cyclic group of prime-cube order	--	$p^{3}$
direct product of cyclic group of prime-square order and cyclic group of prime order	2	1	group of prime order	$p$	SmallGroup(p^4,3), SmallGroup(p^4,4), SmallGroup(p^4,6)	second cohomology group for trivial group action of direct product of cyclic group of prime-square order and cyclic group of prime order on cyclic group of prime order	$p^{4}$
unitriangular matrix group:UT(3,p)	3	2	elementary abelian group of prime-square order	$p^{2}$	lots of them	second cohomology group for trivial group action of UT(3,p) on elementary abelian group of prime-square order	$p^{5}$
semidirect product of cyclic group of prime-square order and cyclic group of prime order	4	2	group of prime order	$p$	lots of them	second cohomology group for trivial group action of semidirect product of cyclic group of prime-square order and cyclic group of prime order on group of prime order	$p^{4}$
elementary abelian group of prime-cube order	5	1	elementary abelian group of prime-cube order	$p^{3}$	lots of them	second cohomology group for trivial group action of elementary abelian group of prime-cube order on elementary abelian group of prime-cube order	$p^{6}$