  Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let
$U(KG)$ denote its group of units. Here without the use of a computer we give
presentations for the unit groups of all group algebras $KG$, where the size of
$KG$ is less than 1024. As a consequence we find the minimum counterexample to
the Isomorphism Problem for group algebras.
