Reference:
Higman, Graham, A finitely generated infinite simple group, J. Lond. Math. Soc. 26, 61-64 (1951). ZBL0042.02201.
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).
