A noncayley graph is a graph which is not a Cayley graph. All graphs that are not vertex-transitive are noncayley graphs. However, some vertex-transitive graph are noncayley. The numbers of vertex-transitive noncayley graphs on n = 1, 2, ... nodes are 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 8, 0, 4, 0, 82, ... (OEIS A006792, McKay and Praeger 1994), with the first nonzero case occurring at n = 10. Royle maintains a list of non-Cayley graphs up to 31 vertices and although the values for 27, 28 and 30 vertices have not been independently verified (though an error in the groups can only affect the graphs if somehow a minimal transitive group has been omitted, so errors are unlikely).

Cayley graph | vertex-transitive graph