I am having trouble construct the following group in GAP. It is a solvable primitive linear group acting on V where |V|=5^8. We know the Fitting subgroup is of order 2^6*4 (central product of extra special group E of order 2^7 with a cyclic group of order 4). On top of E/Z(E) we have a group A of order 6^4 acts on E/Z(E). Here A itself has a normal extra special group D of order 27 and A/D acts on D/Z(D) and A/D \cong GL(2,3). In some sense, G would be a maximal solvable primitive group on V=5^8.
If it is possible, I need similar construction in |V|=7^8.