In this article, we study a class of groups which are commensurable with a direct product of the discrete Heisenberg group and a free abelian group, or a free abelian group by using zeta functions of groups defined by Grunewald, Segal, and Smith as generating functions of the number of subgroups of given index $n$. We will show that zeta functions determine their isomorphism classes for groups belonging to the above class.
"Isomorphism Classes and Zeta-functions of Some Nilpotent Groups." Tokyo J. Math. 36 (1) 163 - 175, June 2013. https://doi.org/10.3836/tjm/1374497517