A space which has an isometry group of $O(n)$ is $S^{n-1}$.
A space which has an isometry group of $U(n)$ is $\mathbb{C}^{n}$ excludes an origin.
Which space has an isometry group of $\operatorname{Spin}(d)$:
When $d$ is even?
When $d$ is odd?
Note that $Spin(d)/(\mathbf{Z}/2\mathbf{Z})=SO(d)$ -- $Spin(d)$ is a universal cover of $SO(d)$.