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.

The MinimumDegreePermutatinoRepresentation( G ) command returns a permutation group isomorphic to the group G and whose degree is minimal, that is, equal to the value returned by MinimumPermutationRepresentationDegree( G ).

You can use the shorter alias MinDegreePermRep'.

In general, these two commands depend upon searching the lattice of subgroups of the group G, so they can be expensive for large groups, and are subject to the limitations of the SubgroupLattice command. The MinPermRepDegree command works for some finite symbolic groups, but the MinDegreePermRep command cannot work for a symbolic group.

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

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

$7$

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

$24$

$\mathrm{MinPermRepDegree}\left(\mathrm{SmallGroup}\left(504\,202\right)\right)$

$19$

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

$8$

$\mathrm{Degree}\left(\mathrm{GU}\left(2\,3\right)\right)$

$32$

$\mathrm{MinPermRepDegree}\left(\mathrm{GU}\left(2\,3\right)\right)$

$24$

$P≔\mathrm{MinDegreePermRep}\left(\mathrm{GU}\left(2\,3\right)\right)$

$P≔⟨\left(1\,18\,14\,13\,3\,19\,9\,17\right)\left(2\,16\,23\,4\,12\,7\,11\,21\right)\left(5\,10\,8\,6\,20\,22\,15\,24\right)\,\left(1\,6\,13\right)\left(2\,15\,16\right)\left(3\,24\,17\right)\left(4\,23\,5\right)\left(7\,12\,8\right)\left(9\,10\,18\right)\left(11\,20\,21\right)\left(14\,22\,19\right)⟩$

$\mathrm{Degree}\left(P\right)$

$24$

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

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

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

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