Cayley's Theorem asserts that each finite group is isomorphic to a group of permutations of a finite set.  In other words, each finite group $G$ can be embedded in a symmetric group $S_n$, for some positive integer $n$.

The MinimumPermutationRepresentationDegree( G ) command returns the minimum degree of a faithful permutation representation for a (finite) group G.  That is the least positive integer n such that G embeds in the symmetric group of degree n.

You can use the alias MinPermRepDegree instead of the longer command name MinimumPermutationRepresentationDegree.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{MinPermRepDegree}\left(\mathrm{CyclicGroup}\left(12\right)\right)$

$7$

$\mathrm{MinPermRepDegree}\left(\mathrm{GL}\left(2\,5\right)\right)$

$24$

$\mathrm{MinPermRepDegree}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$8$

$\mathrm{MinPermRepDegree}\left(\mathrm{PSL}\left(5\,q\right)\right)$

$q^4+q^3+q^2+q+1$

The GroupTheory[MinimumPermutationRepresentationDegree] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.