Reference:

<cite authors="Higman, Graham">_Higman, Graham_, [**A finitely generated infinite simple group**](http://dx.doi.org/10.1112/jlms/s1-26.1.61), J. Lond. Math. Soc. 26, 61-64 (1951). [ZBL0042.02201](https://zbmath.org/?q=an:0042.02201).</cite>    
 
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).