  We compute the generator rank of a subhomgeneous C*-algebra in terms of the
covering dimension of the pieces of its primitive ideal space corresponding to
irreducible representations of a fixed dimension. We deduce that every Z-stable
ASH-algebra has generator rank one, which means that a generic element in such
an algebra is a generator.
  This leads to a strong solution of the generator problem for classifiable,
simple, nuclear C*-algebras: a generic element in each such algebra is a
generator. Examples of Villadsen show that this is not the case for all
separable, simple, nuclear C*-algebras.
