  We establish a congruence satisfied by the integer group determinants for the
non-abelian Heisenberg group of order $p^3$. We characterize all determinant
values coprime to $p$, give sharp divisibility conditions for multiples of $p$,
and determine all values when $p=3$. We also provide new sharp conditions on
the power of $p$ dividing the group determinants for $\mathbb Z_p^2$.
  For a finite group, the integer group determinants can be understood as
corresponding to Lind's generalization of the Mahler measure. We speculate on
the Lind-Mahler measure for the discrete Heisenberg group and for two other
infinite non-abelian groups arising from symmetries of the plane and 3-space.
