The relationship becomes even more fascinating in elementary particle theory. I.e., the insight that the state spaces of elementary (quantum) particles can be constructed by finding representations of a particular group. Also advanced classical mechanics where symmetry groups for the dynamics, and associated structure of the symplectic phase space take center stage. The notion that Minkowski spacetime is "really" just a homogeneous space for the Poincare group is also intriguing.
One might even say that these relationships are now intrinsic to most (if not all) of modern theoretical physics (after one has generalized the concept of "geometry" to "representations").
BTW, what about semigroups? Is there a well developed theory of (some alternate version of) homogeneous spaces when one is dealing with a semigroup in which some elements have no inverses? The obvious example is the heat equation for which only forward time evolution is sensible.
[Edit: Is "Erlanger" a typo? I thought it was "Erlangen".]
There is a very deep link between group theory and geometry.
But Klein had a very general method of generating geometries. Basically, he took projective space and he equipped it with a special "distance functions". Then a lot of very pathological but also natural geometries pop up. For example, of course euclidean, hyperbolic and elliptic geometry shows up this way. But also Minkowski geometry and Galilean geometry shows up in this way outlined by Klein. In this way, Klein discovered Minkowski geometry far before SR and GR, but he probably dismissed it for being not useful.