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I was using an older version of Sage (7.1, I think?) on macOS High Sierra, installed using the binaries on sagemath.org. I've updated my version of Sage and no longer have the TypeError from before, but am not getting the result I'd expect, as elaborated on in the comment below your answer.

I've updated to 8.4 and no longer have this error, but am not getting the expected result:

S = SymmetricGroup(3)
gen1 = Permutation('(1,2)(3)')
gen2 = Permutation('(1,3)(2)')
G = PermutationGroup([gen1])
H = PermutationGroup([gen2])
CCG = ConjugacyClass(S, G)
CCH = ConjugacyClass(S, H)
print CCG == CCH

This prints False, though these groups are conjugate.

If I have two particular subgroups of a symmetric group, is there any built-in way in Sage to determine if the groups are conjugate to one another? I tried creating a ConjugacyClass for each and then comparing them, but this gives an error:

S = SymmetricGroup(3)
gen1 = Permutation('(1,2,3)')
gen2 = Permutation('(1,3,2)')
gen3 = Permutation('(1,2)')
gen4 = Permutation('(1,3)')
G1 = PermutationGroup([gen1, gen3])
G2 = PermutationGroup([gen2, gen4])
ConjugacyClass(S, G1) == ConjugacyClass(S, G2)

When executing the very last line I get the error

TypeError: For implementing multiplication, provide the method '_mul_' for (1,2) resp. Permutation Group with generators [(1,2), (1,2,3)]