**Conjecture:** Let $p$ be a prime.
Then the group 


$G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9, (([a,b]^4)b)^{2p} \rangle$ 


has a composition series of the form
${\rm PSL}(2,8) - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p$.


Is there any literature on this subject, and if not, how can this conjecture be proved?



Group theory conjecture on hurwitz groups
For some reason, people seem to be voting to close this, so I will give a quick reply. Your conjecture is true and it can be proved mainly by computer.


The group $G=\langle a,b \mid a^2, b^3, (ab)^7, [a,b]^9 \rangle$ has a homomorphism onto ${\rm PSL}(2,8)$. Let $K$ be the kernel. Then it can be shown that $K$ is nilpotent of class 2, with $|Z(K)|=2$ and $K/Z(K)$ free abelian of rank 7. The element $x:=([a,b]^4b)^2$ lies in $K$ and maps onto a generator of $K/Z(K)$. So factoring out the normal closure of $x^p$ in $G$ will result in an extension of an elementary abelian group of order $p^7$ by ${\rm PSL}(2,8)$.


