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${\rm GL}(n,p)$ acts primitively (in fact $2$-transitively) on $(p^n-1)/(p-1)$ points, and it has (for $n$ even) an elementary abelian $p$-subgroup of order $p^{n^2/4}$ (think of upper unitriangular matrices with an $n/2 \times n/2$ block in the top right corner), and hence a subgroup chain of length $n^2/4$.