The phase transition of the pure Potts model is indeed first-order for $q\ge 5$ but a continuous transition is induced for any number of states $q$ when the exchange couplings are made random. In the large $q$-limit, the central charge has been conjectured to follow the law

$$c(q)={1\over 2}\ln_2 q$$

See the reference J.L. Jacobsen, and M. Picco (2000), Phys. Rev. E 61, R13. Note that there are some subtleties in the definition of $c$. The precise CFT of the random Potts model is not known but it cannot be unitary.