I am confused with the symmetry group and the representation of spin-$N$ particles, and will appreciate any help or suggestions of reference.

There are $2N+1$ internal states for a (massive) spin-$N$ particle. These internal states define a $2N+1$-dimensional Hilbert space. It seems to be resonable that, the associate symmetry group is $SU(2N+1)$.

However, it seems also possible that, the $2N+1$ states correspond to the $2N+1$-dimension irreducible representation of $SU(2)$.

Are there some relations between the above two situations, or I just confuse some basic concepts?