= Integral central extensions of a rank-two free abelian group
{title2=$E_n=\langle x,y,z:\ z\text{ central},\ [x,y]=z^n\rangle$}
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 $z^my^bx^a$, 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 $H^2(\mathbb Z^2,\mathbb Z)\cong\mathbb Z$.
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