After counting the number of conjugates of $h=(1 2 \ldots n-2)$ I get $n(n-1)(n-3)!$ which when plugged in to the orbit stabilizer theorem gives the centralizer to have order $n-2$ but this would show the opposite of what is asked?
Is it mine or the books mistake?
Never mind. Think I should have divided number of conjugates by two which would show what is required