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when a space (or spacetime) is said to be maximally symmetric, does this mean that it is homogeneous?
Does this mean a flat Euclidean space is more symmetric than spaces with closed and open curvature (hyperspheric and hyperbolic)?
Thus, in addition to a full complement of ordinary Killing vectors, Euclidean space is also blessed with a maximal collection of conformal Killing vectors, making it "uber-maximally symmetric," unlike hyperspheres and hyperboloids. So, indeed, there is a "third kind of invariance," as you say, having to do with changes of scale, that Euclidean space possesses but other so-called maximally symmetric spaces do not.
Noether would say for every symmetry there is an inertia or conservation law, so what is law that results from scale symmetry?