If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{ Z}^3$ then $\beta_1(G;\mathbb{ Q})\leq2$ \cite{ Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{ Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $\beta_1(G;\mathbb{ Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{ Z}$ then $G\cong\mathbb{ Z}/m\mathbb{ Z}\rtimes_n\mathbb{ Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$.
Categories: math.GT