  We study finitely generated nilpotent groups $G$ given by full rank finite
presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of
nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the
deficiency $|A| - |R| $ is at least $2$ then the group $G$ is virtually free
nilpotent, it is quasi finitely axiomatizable (in particular, first-order
rigid), and it is almost (up to finite factors) directly indecomposable. One of
the main results of the paper is that the Diophantine problem in nilpotent
groups given by full rank finite presentations $\langle A \mid R\rangle$ is
undecidable if $|A| - |R| \geq 2$ and decidable otherwise. We show that this
class of groups is rather large since finite presentations asymptotically
almost surely have full rank, so a random nilpotent group in the few relators
model has a full rank presentation asymptotically almost surely. Full rank
presentations give one a useful tool to approach random nilpotent groups and
study their properties. Note, that the results above significantly improve our
understanding of the Diophantine problem in finitely generated nilpotent
groups: from a few special examples of groups with undecidable Diophantine
problem we got to the place where we know that the Diophantine problem in all
"typical" nilpotent groups is also undecidable.
