  We analyse entanglement classes for permutation-symmetric states for n qudits
(i.e. d-level systems), with respect to local unitary operations
(LU-equivalence) and stochastic local operations and classical communication
(SLOCC equivalence). In both cases, we show that the search can be restricted
to operations where the same local operation acts on all qudits, and we provide
an explicit construction for it. Stabilizers of states in the form of
one-particle operations preserving permutation symmetry are shown to provide a
coarse-grained classification of entanglement classes. We prove that the Jordan
form of such one-particle operator is a SLOCC invariant. We find, as
representatives of those classes, a discrete set of entangled states that
generalize the GHZ and W state for the many-particle qudit case. In the later
case, we introduce "excitation states" as a natural generalization of the W
state for d>2.
