The component groups $\pi_0(C_G(X))$ are known by work of Alekseevskiĭ, [MR557505][1] = [MR2140712][2], which contains additional references and can be read on [Google books][3]. Another good place to look is the book by Collingwood & McGovern, [*Nilpotent orbits in semisimple Lie algebras*][4].

(Out of curiosity, what other topological invariants are you after?)

  [1]:http://www.ams.org/mathscinet-getitem?MR=557505
  [2]:http://www.ams.org/mathscinet-getitem?MR=2140712
  [3]:http://books.google.com/books?id=6VqKgvUTncoC&pg=PA15
  [4]:http://www.ams.org/mathscinet-getitem?MR=1251060