  Let $\Gamma$ denote a central extension of the form $1\to
\mathbb{Z}^r\to\Gamma\to \mathbb{Z}^n\to 1$. In this paper we describe the
topology of the spaces of homomorphisms $\text{Hom}(\Gamma, U(m))$ and the
associated moduli spaces $\text{Rep}(\Gamma, U(m))$, where $U(m)$ is the group
of $m\times m$ unitary matrices.
