  We investigate the critical properties of a one dimensional stochastic
lattice model with n (permutation symmetric) absorbing states. We analyze the
cases with $n \leq 4$ by means of the non-hermitian density matrix
renormalization group. For n=1 and n=2 we find that the model is respectively
in the directed percolation and parity conserving universality class,
consistent with previous studies. For n=3 and n=4, the model is in the active
phase in the whole parameter space and the critical point is shifted to the
limit of one infinite reaction rate. We show that in this limit the dynamics of
the model can be mapped onto that of a zero temperature n-state Potts model. On
the basis of our numerical and analytical results we conjecture that the model
is in the same universality class for all $n \geq 3$ with exponents $z =
\nu_\|/\nu_\perp = 2$, $\nu_\perp = 1$ and $\beta = 1$. These exponents
coincide with those of the multispecies (bosonic) branching annihilating random
walks. For n=3 we also show that, upon breaking the symmetry to a lower one
($Z_2$), one gets a transition either in the directed percolation, or in the
parity conserving class, depending on the choice of parameters.
