  We investigate the minimum distance of the error correcting code formed by
the homomorphisms between two finite groups $G$ and $H$. We prove some general
structural results on how the distance behaves with respect to natural group
operations, such as passing to subgroups and quotients, and taking products.
Our main result is a general formula for the distance when $G$ is solvable or
$H$ is nilpotent, in terms of the normal subgroup structure of $G$ as well as
the prime divisors of $|G|$ and $|H|$. In particular, we show that in the above
case, the distance is independent of the subgroup structure of $H$. We
complement this by showing that, in general, the distance depends on the
subgroup structure $G$.
