Let $G$ be a finite abelian group of rank $n$ and $H\rightarrow G$ a central extension with cyclic kernel. Is it true that we can find a faithful representation $H\rightarrow {\rm GL}_{k(n)}(\mathbb{C})$ where $k(n)$ only depends on $n$?

I feel something like this must be true from the fact that $G$ should admit an irreducible projective representation into ${\rm PGL}_{k(n)}(\mathbb{C})$ where $k(n)$ only depends on $n$.