Solution
= Solution
A <Sylow subgroup> for $2$ has order eight. Every one of its nonidentity elements lies in the only even-order class, so has order two. In a <group> of exponent two,
$$
xy=(xy)^{-1}=y^{-1}x^{-1}=yx.
$$
It is therefore an <elementary abelian group> and $\boxed{S\cong C_2^3}$.