The free group on n generators modulo n plus u random relations as n goes to infinity

We show that, as n goes to infinity, the free group on n generators, modulo n + u random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen-Lenstra heuristics. For each n, these random groups belong to the few relator model in the Gromov model of random groups.