A balanced presentation for a group is a presentation with an equal number of generators and relations. A perfect group is a group that is equal to its commutator group.
I am wondering what finite perfect groups are known to admit balanced presentations? I only know two example: the trivial group and the binary icosahedral group $\langle x,y \mid x^2=y^3=(xy)^5\rangle$. What are some other examples?