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?