Groups of order 250

This article gives information about, and links to more details on, groups of order 250
See pages on algebraic structures of order 250 | See pages on groups of a particular order

Statistics at a glance

The number 250 has prime factors 2 and 5. The prime factorization is as follows:

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

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

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

The order 250 is part of GAP's SmallGroup library. Hence, any group of order 250 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 250 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

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.