Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group of rank $r$ and step $n$.
Consider now an arbitrary nilpotent group $N$ of rank $r$. I am asked to show that $N \cong F_{n,r}/K$ for some $K \triangleleft F_{n,r}$.
I am thinking that perhaps I am meant to consider all possible word expressions of elements in $N$ in terms of some generating set $S$ of rank $n$, and then show that the things I'd want to quotient out by, when thought of as elements of $F_{n,r}$, form a normal subgroup.
Another idea I had was that I could possibly show that a general, not-necessarily nilpotent, is isomorphic to a quotient of $F^{(r)}$, but that has the similar problem of, I'm not sure how to characterise all expressions that I want to "quotient" out by.
These problems make me think that perhaps I should approach this from a different point of view. These approaches are direct in the direction of "given such an $N$, I want to find an appropriate $K$".
However, I am not really sure how else I can go about this. Perhaps there is a proof by contradiction where I may assume I have a nilpotent group that is not isomorphic to any quotient of the free nilpotent group, but then in that case I can't anticipate what exactly I would be contradicting.
I would really appreciate any advice regarding what method of proof may yield the best results for me in regards to this problem, and also how I might go about starting it if it requires some inspiration. Thank you!