What is the effect of the composition $f ∘ g ∘ f^{−1}$ on g?

Question

Let f = (4 5 6) and g = (1 9 8 4)(2 7 5)(3 6) be two permutations in S₉.

Then $f ◦ g ◦ f^{−1}$ = (1 9 8 5)(2 7 6)(3 4)

What is the effect of such a composition on g?

I can see 2 potential similarities. The first one is that the 4 5 and 6 have been replaced by the image of $f$. The second has nothing to do with $f$ and the end values in each cycle for $g$ has been pushed once. I'm really not too sure.

I can see a similarity, 4 5 and 6 have been replaced by the image of f but then I'm doubting myself and thinking the actual values in f have no effect on g.

For example, how would this look like in the follow proposition:

Let n be a positive integer and let f,g ∈ S_n. Suppose that the cycle notation for g is (a_1,1 a_1,2 ··· a_1,k1 )···(a_m,1 a_m,2 ···a_m,km ). Then the cycle notation for $f ◦ g ◦ f^{−1}$ is ...

Any help is much appreciated.

See eg https://math.stackexchange.com/questions/120971/conjugate-permutations-in-s-n-and-or-a-n — ancient mathematician, Apr 14 '23 at 08:27
@ancientmathematician thanks but I'm still having a hard time finding what conjugate means in regards to my context. could you explain a bit further sorry for the hassle — Bill Cogn, Apr 14 '23 at 08:31
Your first idea is good, but the statement can go further. Does this replacement only apply to $4,5,6$? — Michael Burr, Apr 14 '23 at 08:31
@MichaelBurr No, do the rest of the values in f that completes S9 also apply? So 1 1, 2 2 etc.. — Bill Cogn, Apr 14 '23 at 08:34
So f◦g◦f−1 is conjugate to g since f,f-1 exists but how would you put that into the proposition? — Bill Cogn, Apr 14 '23 at 08:36
@MichaelBurr Wait I think I got it. Is the cycle notation for f g f-1 = (f(a1,1),f(a,1,2)...)? — Bill Cogn, Apr 14 '23 at 08:41
You can make up your own example to see if your formula works. — Michael Burr, Apr 14 '23 at 08:44
@MichaelBurr Yes it works wow. That's really interesting to see how that works. Intuitively, how is f involved in this? — Bill Cogn, Apr 14 '23 at 08:46
If you apply $f\circ g\circ f^{-1}$ to $f(x)$, then you get $f(g(x))$ — Michael Burr, Apr 14 '23 at 09:12

What is the effect of the composition $f ∘ g ∘ f^{−1}$ on g?

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