elements of symmetric groups and centralizers

Let P = (1, 2, 3, .... n) in $\displaystyle S_n$.

a. Show there are (n-1)! distinct n-cycles in $\displaystyle S_n$.

b. How many conjugates does P have in $\displaystyle S_n$?

c. Let C(P)=N(P) be the centralizer of P in $\displaystyle S_n$. Using (b), find |C(P)|.

d. Find the subgroup C(P).

my thoughts:

a -- this seems really obvious, but I never actually learned how to show it! I have no ideas.

b. by part (a), wouldn't this be simply (n-1)!, since that's the property of symmetric groups?

c & d, I really don't know what to do. Do I use the class equation somehow, or another theorem?

Thanks so much for any help!