A compact, connected Lie group is isomorphic to the quotient of a torus $T$ and a simply-connected, compact Lie group $J$ by a finite central subgroup $Z\subseteq T\times J$: $$G\simeq (T\times J)/Z.$$
Main Questions: Are we able to tell anything about $\pi_1(G)$ from $Z$ when $T$ is not the trivial group? Conversely, if $\pi_1(G)=\mathbb{Z}^n$, does that mean that we can take $Z=1$?
Work on Problem: I know that if $G$ is a semi-simple Lie group ($T=1$), then $G\simeq J/Z$. Using the homotopy exact sequence for the fibration $J\rightarrow G$ with fibre $Z$, we have $$\pi_1(J)\rightarrow \pi_1(G)\rightarrow \pi_0(Z)\rightarrow \pi_0(J).$$ As $J$ is simply-connected, we have $\pi_1(G)\simeq \pi_0(J)\simeq \mathbb{Z}_{|Z|}$.
If we do the same thing for $T$ general, we have the exact sequence $$\pi_1(Z)\rightarrow \pi_1(T\times J)\rightarrow \pi_1(G)\rightarrow \pi_0(Z)\rightarrow \pi_0(T\times J).$$ Substituting in the known homotopy groups, we have $$1\rightarrow \mathbb{Z}^{\mathrm{dim}(T)}\rightarrow\pi_1(G)\rightarrow \mathbb{Z}_{|Z|}\rightarrow 1.$$ As $G$ is a compact, connected Lie group, $\pi_1(G)$ is a finitely generated abelian group. While this tells me the rank of $G$, how can I determine the torsion of $G$?