Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the bordism group of $BG$ where $BG$ is the classifying space of a group $G$. The $d$-th bordism group is $\Omega_d^H (BG)$.
Now let us consider the quotient of their classifying spaces $BSU(3)/BSO(3)$ (either the point set quotient or the mapping cone).
I am not aware of the bordism group of $$\Omega_d^H(BSU(3)/BSO(3))$$ of this space $BSU(3)/BSO(3)$. Are there previous knowledge of computing the bordism group of this type?
Here $H$ can be some structures equipped for the cobordant of manifolds.