On Heisenberg groups

It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ 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 $G/[G,G]$ to a Heisenberg group ${\mathscr{H}}(G/[G,G], [G,G], c')$ whose $2$-cocycle $c'$ is bimultiplicative.