Following this, we consider a more advanced question, below we take $N=3$ for all $N$.
- Consider the group:
$$\mathcal{G}=\frac{SU(N)_A \times SU(N)_{B_1} \times SU(N)_{B_2} \times U(1)}{(\mathbb{Z}_N)^2},$$
this group can be understood as a 4-multiplet $$(g_A, g_{B,1},g_{B,2}, e^{i \theta}) \in SU(N)_A \times SU(N)_{B_1} \times SU(N)_{B_2}\times U(1),$$ such that $(e^{i \frac{2\pi}{N}}, 1, 1, e^{-i \frac{2\pi}{N}})$ is the first $\mathbb{Z}_N$ generator mod out in $G$, while $(1,e^{i \frac{2\pi}{N}},e^{i \frac{2\pi}{N}}, e^{-i \frac{2\pi}{N}})$ is the second $\mathbb{Z}_N$ generator mod out in $G$. Namely (1) the center of $SU(N)_A$, (2) the center of $SU(N)_{B,1}$ together with the center of $SU(N)_{B,2}$, and (3) the $e^{i \frac{2\pi}{N}}\in U(1)$ overlap, thus we only mod out the twice redundant $(\mathbb{Z}_N)^2$.
Now consider the subgroup $H$ of the $G$ as:
$$H=\frac{SU(N)_{A,B}}{\mathbb{Z}_N}\times \mathbb{Z}_2,$$
where this group can be understood as a doublet $(g_{A,B}, g_1) \in SU(N)_{A,B}\times \mathbb{Z}_2= (SU(N)_{A,B}, \{\pm 1 \}) =H \subset G,$ where there is a one-to-one correspondence between the doublet $$(g_{A,B}, g_1) \in H$$ and the 4-multiplet $$(g_{A,B},g_{A,B}^*,g_{A,B}^*, g_1) \in G$$ with $g_{A,B}=g_A=g_{B,1}^*=g_{B,2}^*$, where $g^*$ means the complex conjugation (without the transpose $T$) of $g$.
Finally, we need $\frac{SU(N)_{A,B}}{\mathbb{Z}_N}$ to mod out ${\mathbb{Z}_N}$, where we consider $(g_{A,B}',g_{A,B}'^*)=(e^{i \frac{2\pi}{N}},e^{-i \frac{2\pi}{N}})\mathbb{I} \in {\mathbb{Z}_N}$; this particular rank-$N$ diagonal matrix is the ${\mathbb{Z}_N}$ we modded out .
We have that $$\frac{SU(N)_{A,B}}{\mathbb{Z}_N} \subset SU(N)_A \times SU(N)_{B_1} \times SU(N)_{B_2} , $$ $$ \mathbb{Z}_2 = \{\pm 1 \} \subset U(1). $$
So this explains how $H$ is embedded as a subgroup in $G$.
Question:
What is the precise space of the set of left coests $$ G/H=? $$ Is this certain smooth homogeneous space like a sphere or a complex/real projective space?
What is the homotopy group? $$ \pi_j(G/H)=? $$ for $j=1,2,3,4,5$.
Some background info that I prepared for you:
(1). $\pi_i(U(N))=\pi_i(\frac{SU(N)\times U(1)}{\mathbb{Z}_N})$: $$\pi_m(U(N))=\pi_m(SU(N)), \text{ for } m \geq 2$$ $$\pi_1(U(N))=\mathbb{Z}, \;\;\pi_1(SU(N))=0,$$
(2). $\pi_1(\mathbb{Z}_N)=\mathbb{Z}_N$ and $\pi_i(\mathbb{Z}_N)=0$ for $i\geq 2$.