  There is a one-to-one correspondence between involutory latin quandles and
uniquely $2$-divisible Bruck loops. Bruck loops of odd prime power order are
centrally nilpotent. Using linear-algebraic approach to central extensions, we
enumerate Bruck loops (and hence involutory latin quandles) of order $3^k$ for
$k\le 5$, except for those loops that are central extensions of the cyclic
group of order $3$ by the elementary abelian group of order $3^4$.
  Among the constructed loops there is a Bruck loop of order $3^5$ whose
associated $\Gamma$-loop is not a commutative automorphic loop. We
independently enumerate commutative automorphic loops of order $3^k$ for $k\le
5$, with the same omission as in the case of Bruck loops.

Group Theory