  Moddings by cyclic permutation symmetries are performed in 4-dimensional
strings, built up from N=2 coset models of the type $CP_m=SU(m+1)/SU(m)\times
U(1)$. For some exemplifying cases, the massless chiral and antichiral states
of $E_6$ are computed. The extent of the equivalence between different
conformal invariant theories which possess equal chiral rings is analyzed.
