  We study representation zeta functions of finitely generated, torsion-free
nilpotent groups which are rational points of unipotent group schemes over
rings of integers of number fields. Using the Kirillov orbit method and p-adic
integration, we prove rationality and functional equations for almost all local
factors of the Euler products of these zeta functions. We further give explicit
formulae, in terms of Dedekind zeta functions, for the zeta functions of
class-2-nilpotent groups obtained from three infinite families of group
schemes, generalising the integral Heisenberg group. As an immediate corollary,
we obtain precise asymptotics for the representation growth of these groups,
and key analytic properties of their zeta functions, such as meromorphic
continuation. We express the local factors of these zeta functions in terms of
generating functions for finite Weyl groups of type B. This allows us to
establish a formula for the joint distribution of three functions, or
'statistics', on such Weyl groups. Finally, we compare our explicit formulae to
p-adic integrals associated to relative invariants of three infinite families
of prehomogeneous vector spaces.
