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. 



What is the simplest known finite presentation of a free nilpotent group?
Taking "simplest" to mean "having fewest generators", the question is how many generators you need to normally generate $\Gamma\_n$ in a free group. As mentioned already in the comments, $\Gamma\_n / \Gamma\_{n+1}$ is free abelian with rank given by Witt's formula, so certainly you need at least this many generators. It seems to be unknown (!) whether the basic commutators of weight $n$ normally generate $\Gamma\_n$. A couple of references I found:


**Sims, Charles C.** Verifying nilpotence. J. Symbolic Comput. 3 (1987), no. 3, 231--247. <https://doi.org/10.1016/S0747-7171(87)80002-0>


**Jackson, David A.** Basic commutators in weights six and seven as relators. Comm. Algebra 36 (2008), no. 8, 2905--2909. <https://doi.org/10.1080/00927870802108148>


