I gave the following GAP-Commands :
gap> a:=(1,2)(3,5)(4,6)(7,10);;
gap> b:=(2,3,4)(5,7,8)(6,9,10);;
gap> G:=Group(a,b);;
gap> PermGroupOps.ElementProperty(G,g->a^g=a);
()
gap> C:=Centralizer(G,a);
Subgroup( Group( ( 1, 2)( 3, 5)( 4, 6)( 7,10), ( 2, 3, 4)( 5, 7, 8)
( 6, 9,10) ), [ ( 3, 4)( 5, 6)( 7,10)( 8, 9), ( 1, 2)( 3, 5)( 4, 6)( 7,10) ] )
gap> c1:=C.generators[1];;
gap> c2:=C.generators[2];;
gap> a^c1;
( 1, 2)( 3, 5)( 4, 6)( 7,10)
gap> a^c2;
( 1, 2)( 3, 5)( 4, 6)( 7,10)

I think that the command PermGroupOps.ElementProperty should have returned
one of the two generators of C or at least a product of them.
Is there an Error or do I make anything wrong ?

Saludos,
        Ansgar