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 ${\pi }_1^{\mathrm{t}\mathrm{o}\mathrm{p}}(X(\mathbb{C}),x)$. Moreover, we compute the second homotopy group ${\pi }_2^{\mathrm{t}\mathrm{o}\mathrm{p}}(X(\mathbb{C}),x)$.