Dear Mathias,

> in the online-manual of GAP 3.4.4 one can read
> 
>  'ConjugacyClass( <G>, <g> )'
>     
>  'ConjugacyClass' returns the conjugacy class of the element <g> in 
>  the group <G>. Signals an error if <g> is not an element in <G>.
>                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> 
> Let's try and see if this is the truth:
Fortunately, the manual does *not* tell the truth here, as your example 
shows. 

> 
>   gap> G:=SymmetricGroup( 5 );; G.name:="S5";;
>   gap> (1,6) in Elements( G );
>   false
>   #
>   # Only to be sure; and now ...
>   #
>   gap> c := ConjugacyClass( G, (1,6) ); 
>   ConjugacyClass( S5, (1,6) )
>   gap> Elements( c );
>   [ (5,6), (4,6), (3,6), (2,6), (1,6) ]
>   #
>   # Oops! Looks like something went quite wrong here :-(
>   #
Everything goes as I would expect in this example! (1,6) is a
permutation on which the elements of G can operate by conjugation. 
So 'ConjugacyClass( G, (1,6) )' should be the set of permutations
   { x^(-1) * (1,6) * x  |  x in G }
and that is exactly what it is:

gap> Set(List(Elements(G), x-> (1,6)^x))=Elements(c); 
true

However, if you want to know in which conjugacy class of S5 the
"element" (1,6) lies GAP 3.4.4 tells you

gap> classes := ConjugacyClasses( G );
[ ConjugacyClass( S5, () ), ConjugacyClass( S5, (1,2) ), 
  ConjugacyClass( S5, (1,2)(3,4) ), ConjugacyClass( S5, (1,2,3) ), 
  ConjugacyClass( S5, (1,2,3)(4,5) ), ConjugacyClass( S5, (1,2,3,4) ), 
  ConjugacyClass( S5, (1,2,3,4,5) ) ]
gap> PositionProperty( classes, x->x=ConjugacyClass(G,(1,6)) ); 
false

I'm happy that GAP detects that c is not equal to any (inner)
conjugacy class of G.

(I said above that "fortunately" the manual is wrong because I'm using
the feature you showed in your example in my programs.)

Best regards,
Frank Luebeck