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).