Solution
= Solution
Writing $D_8=\langle r,s:r^4=s^2=e, srs=r^{-1}\rangle$, one has $[r,s]=r^{-2}=r^2$. The quotient by $\langle r^2\rangle$ is $C_2\times C_2$, so
$$
\boxed{D_8'=\{e,r^2\}.}
$$
= Solution
Writing $D_8=\langle r,s:r^4=s^2=e, srs=r^{-1}\rangle$, one has $[r,s]=r^{-2}=r^2$. The quotient by $\langle r^2\rangle$ is $C_2\times C_2$, so
$$
\boxed{D_8'=\{e,r^2\}.}
$$