Integral central extensions of a rank-two free abelian group ($E_n=⟨x,y,z:z\text{ central},[x,y]=z^n⟩$)

With a fixed identification of the kernel with $\mathbb{Z}$ and quotient with ${\mathbb{Z}}^2$, these central group extensions are classified by $n\in \mathbb{Z}$. Choose lifts $x,y$ of the quotient generators; their commutator is $z^n$ and is unchanged by central changes of lifts. Every element has unique normal form $z^m{y}^b{x}^a$, proving this invariant is complete. A cocycle representative is $c((a,b),(c,d))=nad$, whose cocycle identity follows from bilinearity. Its extension multiplication is $(a,b,m)(c,d,l)=(a+c,b+d,m+l+nad)$. The zero parameter gives ${\mathbb{Z}}^3$ and parameter one gives the integer Heisenberg group. The Koszul resolution for a rank-two free abelian group independently computes $H^2({\mathbb{Z}}^2,\mathbb{Z})\cong \mathbb{Z}$.