  We initiate study of the Terwilliger algebra and related semidefinite
programming techniques for the conjugacy scheme of the symmetric group
Sym$(n)$. In particular, we compute orbits of ordered pairs on Sym$(n)$ acted
upon by conjugation and inversion, explore a block diagonalization of the
associated algebra, and obtain improved upper bounds on the size $M(n,d)$ of
permutation codes of lengths up to 7. For instance, these techniques detect the
nonexistence of the projective plane of order six via $M(6,5)<30$ and yield a
new best bound $M(7,4) \le 535$ for a challenging open case. Each of these
represents an improvement on earlier Delsarte linear programming results.
