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CycleIndexPolynomial[Cycles[{}], {Subscript[x, 1]}]

returns 1.

I was expecting Subscript[x, 1].

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You have to specify the size of the set that the group is acting on. For the regular representation, the set that the group is acting on is the group itself, and so you have to specify the size of the group. CycleIndexPolynomial takes a third argument where you can specify the size of the set that the group acts on. So, if your group is of size 8, do

CycleIndexPolynomial[Cycles[{}], Array[x, 1], 8]
(* x[1]^8 *)

In the case where your group is the group of size 1, do

CycleIndexPolynomial[Cycles[{}], Array[x, 1], 1]
(* x[1] *)

Mathematica cannot know how large the group is (or how large the set that the group is acting on is), so you have to tell it. The default behavior is perhaps a little strange, but it needs to spit out something. Ideally, it would spit out an error, but Mathematica's group theory functionality is a little bit clunky at times.

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