Groups of order 250

Statistics at a glance
The number 250 has prime factors 2 and 5. The prime factorization is as follows:

$$\! 250 = 2 \cdot 5^3 = 2 \cdot 125$$

In fact, we can say more: the number of 5-Sylow subgroups for any group of order 250 is 1, so we have a normal Sylow subgroup for the prime 5 (of order 125) with a complement of order two which is the 2-Sylow subgroup. The group is thus an internal semidirect product of its 5-Sylow subgroup by the action of its 2-Sylow subgroup. There are thus two possibilities:


 * The group is a finite nilpotent group: It is a direct product of its 5-Sylow subgroup and its 2-Sylow subgroup.
 * The group is a semidirect product of a group of order $$5^3$$ by the action of a non-identity automorphism of order two.

GAP implementation
gap> SmallGroupsInformation(250);

There are 15 groups of order 250. They are sorted by their Frattini factors. 1 has Frattini factor [ 10, 1 ]. 2 has Frattini factor [ 10, 2 ]. 3 - 6 have Frattini factor [ 50, 3 ]. 7 - 8 have Frattini factor [ 50, 4 ]. 9 - 11 have Frattini factor [ 50, 5 ]. 12 - 15 have trivial Frattini subgroup.

For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.