We study the genus and SNT sets of connective covering spaces of familiar finite CW-complexes, both of rationally elliptic type (e.g. quaternionic projective spaces) and of rationally hyperbolic type (e.g. one-point union of a pair of spheres). In connection with the latter situation, we are led to an independently interesting question in group theory: if f is a homomorphism from Gl(ν,A) to Gl(n,A), ν < n, A = ℤ, resp. $ℤ_p$, does the image of f have infinite, resp. uncountably infinite, index in Gl(n,A)?