The component groups $\pi_0(C_G(X))$ are known by work of Alekseevskiĭ, MR557505 = MR2140712, which contains additional references and can be read on Google books. Another good place to look is the book by Collingwood & McGovern, Nilpotent orbits in semisimple Lie algebras.
(Out of curiosity, what other topological invariants are you after?)