 At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete.This is the case for all of the symmetric groups on elements when is not 2 or 6.When , the symmetric group has a unique non-trivial class of outer automorphisms, and when , the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.