The first and second homotopy groups of a homogeneous space of a complex linear algebraic group

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we compute the topological fundamental group $\pi_1^{\rm top}(X({\mathbb C}),x)$. Moreover, we compute the second homotopy group $\pi_2^{\rm top}(X({\mathbb C}),x)$.
Comment: V.1: 10 pages. V.2: 13 pages. V.3: 13 pages. This corresponds to the final version to appear in J. Lie Theory, except that we have added a reference to a very recent preprint of Popov