Reference:
Higman, Graham
A finitely generated infinite simple group. J. London Math. Soc. 26, (1951). 61--64
It is shown that G is infinite and has no proper normal subgroups of finite index, except G.
It is easy to see that this group is perfect: it has trivial abelianization.
I have heard through the grapevine that the space $X$ with four one-cells and four two-cells (corresponding, respectively, to generators and relations) is the classifying space of the group (I don't have a reference).