Free nilpotent group

The free nilpotent group of rank $r$ and class $c$ is $F_r/{\gamma }_{c+1}(F_r)$, where $F_r$ is the free group of rank $r$ and ${\gamma }_i$ is its lower central series.

The quotient mapis a central nonsplit extension of the Integer Heisenberg group by ${\gamma }_3(F_2)/{\gamma }_4(F_2)\cong {\mathbb{Z}}^2$. It cannot split because the source has abelianization ${\mathbb{Z}}^2$, whereas a central splitting would give an abelianization containing an additional direct factor ${\mathbb{Z}}^2$.