Computing homotopy group $\pi_d\left(\mathrm{SO}(2N)/\mathrm{SO}(N) \right)$

Question

I am not a mathematician, but am interested in computing this homotopy group. I have two questions related to this.

1. How can I use the long exact sequence to compute this homotopy group? Part of it would look like $\pi_n\left(\mathrm{SO}(N) \right) \rightarrow \pi_n\left(\mathrm{SO}(2N) \right) \rightarrow \pi_n\left(\mathrm{SO}(2N)/\mathrm{SO}(N)\right)$, I think. How can I use this to compute the homotopy group of interest?

2. Can I compute this homotopy group by finding a homomorphism between $\mathrm{SO}(2N)/\mathrm{SO}(N)$ and $\mathrm{SO}(2N)/\mathrm{U}(N)$, to show that $\pi_d\left(\mathrm{SO}(2N)/\mathrm{SO}(N) \right)$ is isomorphic to $\pi_d\left(\mathrm{SO}(2N)/\mathrm{U}(N) \right)$? This relates to the following proofwiki page: https://proofwiki.org/wiki/Homotopy_Group_is_Homeomorphism_Invariant.