Having found that a group G has a normal Sylow 2-subgroup P, how do I find $C_P(g_i)$, where $g_i$ is a conjugacy class representative?
I have the character table, and have previously found $|C_G(g_i)|$ and shown that $G/[G,G]$ is cyclic, if those results are relevant.
Thanks in advance!
Edit: $g_i$ is the cyclic group of order 5 $\left\{1, a, a^{2}, a^3, a^4 \right\}$ where $a^5 = 1$.