Frank Luebeck writes, in response to Mathias Kratzer's message:

Fortunately, the manual does *not* tell the truth here, as your example
shows.
(I said above that "fortunately" the manual is wrong because I'm using
the feature you showed in your example in my programs.)

The point appears indeed to be that GAP does not check whether g is in G or
not when setting up
ConjugacyClass (G, g);
In fact
Print(GroupOps.ConjugacyClass);
yields

function ( G, g )
    local  C;
    C := rec(
         );
    C.isDomain := true;
    C.isConjugacyClass := true;
    C.group := G;
    C.representative := g;
    C.operations := ConjugacyClassGroupOps;
    return C;
end

When you ask for the elements, the orbit of g under the action of G is
constructed. The nature of g is immaterial. This can lead to the following:

S5 := SymmetricGroup (5);
c := ConjugacyClass (S5, 1);;
Elements (c);

Andreas Caranti