Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 2 / a / 1 / Solution

The Integer Heisenberg groupis generated by the matrices with $(a,b,c)=(1,0,0)$ and $(0,1,0)$. Their group commutator is the nonidentity central matrix with $(a,b,c)=(0,0,1)$. Thus it is nonabelian and nilpotent of class two, while being finitely generated.