On Heisenberg groups

It is known that an abelian group $A$ and a $2$-cocycle $c:A\times A\to C$ yield a group $\mathcal{H}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In this note, we prove that under mild conditions, any class~$2$ nilpotent group $G$ is equivalent as an extension of $G\mathrm{/}[G,G]$ to a Heisenberg group $\mathcal{H}(G\mathrm{/}[G,G],[G,G],c^{\mathrm{\prime }})$ whose $2$-cocycle $c^{\mathrm{\prime }}$ is bimultiplicative.