Solution
= Solution
The <Integer Heisenberg group>
$$
H_3(\mathbb Z)=
\left\{
\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}:a,b,c\in\mathbb Z
\right\}
$$
is 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 group>[nilpotent] of class two, while being finitely generated.