There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation pi of G such that the image pi(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image pi(L1(G)) of the L1-group algebra does not contain any nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a ''generalized Heisenberg group''.