Interesting finite groups tend to have interesting inherent geometries (just as orbit-stabilizer turns external actions into internal actions, similar ideas turn many external geometries into coset geometries). The geometry induced by conjugation on Sylow p-subgroups is important for all finite groups, and turns out to describe the (p-completion of the) classifying space of the group.
Geometry has always been an important part of group theory. Zassenhaus groups and sharply triply transitive groups typically have an underlying affine or projective plane they are acting on. Early investigations of these special permutation groups in the 1930s led to some of the systematic development of finite geometry over things other than fields. You can recover the algebraic structure of something like a ring just from the permutation action of the group (often on a regular subgroup). M. Hall Jr.'s textbook on the Theory of Groups has a nice exposition of these ideas.
Of course finite groups of Lie type acting on their Borel subgroups also define important geometries, roughly called "buildings", and there are a great many references for those. This became a very popular way to understand the non-sporadic groups. These groups of Lie type have other nice actions, often on interesting finite geometries called generalized polygons.
Equivariant homotopy people noticed that some of these geometries are nearly enough to define a classifying space of the group, along with a nice decomposition of its cohomology ring. D. Benson and S.D. Smith's book on Classifying Spaces of Sporadic Simple Groups (MR2378355) describes these techniques with a reasonably algebraic feel. Modulo a few details, these are the fusion systems Scott Carnahan mention in a previous thread, MO5659. These geometries were investigated in order to provide a more natural analogue of buildings for sporadic groups.
Actually, I suppose you might feel that classifying spaces themselves are naturally associated to finite groups.