Searching for small simple automorphic loops

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no nonassociative simple commutative automorphic loop of order less than $2^{12}$, and no nonassociative simple automorphic loop of order less that 2500. We obtain examples of nonassociative simple right automorphic loops.