  We show that if a torsion free nilpotent group $G$ has a balanced
presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There
is just one such group which is torsion-free and of Hirsch length $h=4$, and
none with $h=5$. We also construct a torsion-free nilpotent group $G$ with
$h=6$ and such that $\beta_2(G;F)=\beta_1(G;F)$ for all fields $F$.
