Rank of a transitive permutation group ($\mathrm{rank}(G)=|G_{\alpha }\backslash \Omega |$)

The number of point-stabilizer orbits, equivalently the number of orbits on ordered pairs. On $k$-spaces of an $n$-space with $k\le n/2$, $G{L}_n(q)$ has rank $k+1$, indexed by intersection dimension.