Finitely generated nilpotent groups are always finitely presented. This is true for abelian groups, and can be shown by induction for nilpotent ones, using the classical lift of a presentation of $N$ and a presentation of $G/N$ to a presentation of $G$. As a consequence, the free $c$-nilpotent group of rank $n$, $F_n/\Gamma_{c+1}(F_n)$, is finitely presented. In fact, I do know a finite presentation : kill all iterated commutators of the generators of length between $c+1$ and $2c$. This works because all iterated commutators of length greater than $c$ can be written as a product of iterated commutators of the generators of length greater than $c$ (using basic commutator calculus), and every such commutator has a sub-commutator of length between $c+1$ and $2c$ (it can help to think of commutators as rooted planar binary trees).

But I am unable to find a simpler presentation (say, with less relations). In fact, I can think of many presentations making the lower central series to stop, and giving the right Lie algebra (using linear trees, or Lyndon trees), but I do not know how to show that the result is nilpotent, and it may well not be. And I have not been able to find a simpler presentation from the induction process described above, either. Whence my question.