The 2-Closure of a 32 -Transitive Group in Polynomial Time

Let G be a permutation group on a finite set Ω. The k-closure G ^(k) of G is the largest subgroup of the symmetric group Sym(Ω) having the same orbits with G on the kth Cartesian power Ω ^k of Ω. The group G is called 32-transitive, if G is transitive and the orbits of a point stabilizer G _α on Ω{α} are of the same size greater than 1. We prove that the 2-closure G ⁽²⁾ of a 32-transitive permutation group G can be found in polynomial time in size of Ω. Moreover, if the group G is not 2-transitive, then for every positive integer k its k-closure can be found within the same time. Applying the result, we prove the existence of a polynomial-time algorithm for solving the isomorphism problem for schurian 32-homogeneous coherent configurations, that is coherent configurations naturally associated with 32-transitive groups.