# Groups of order 320

## Contents

See pages on algebraic structures of order 320| See pages on groups of a particular order

## Statistics at a glance

The number 320 has prime factorization $320 = 2^6 \cdot 5$. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 320 are solvable groups (specifically, finite solvable groups).

Quantity Value Explanation
number of groups up to isomorphism 1640
number of abelian groups up to isomorphism 11 (number of abelian groups of order $2^6$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 6) times (number of unordered integer partitions of 1) = $11 \times 1 = 11$. See also classification of finite abelian groups
number of nilpotent groups up to isomorphism 267 (number of groups of order 64) times (number of groups of order 5) = $267 \times 1 = 267$
number of solvable groups up to isomorphism 1640 There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 320 are solvable groups (specifically, finite solvable groups).
number of simple groups up to isomorphism 0 All groups of this order are solvable, so there cannot be any simple groups.

## GAP implementation

The order 320 is part of GAP's SmallGroup library. Hence, any group of order 320 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 320 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(320);

There are 1640 groups of order 320.
They are sorted by their Frattini factors.
1 has Frattini factor [ 10, 1 ].
2 has Frattini factor [ 10, 2 ].
3 has Frattini factor [ 20, 3 ].
4 - 125 have Frattini factor [ 20, 4 ].
126 - 178 have Frattini factor [ 20, 5 ].
179 - 272 have Frattini factor [ 40, 12 ].
273 - 874 have Frattini factor [ 40, 13 ].
875 - 1011 have Frattini factor [ 40, 14 ].
1012 has Frattini factor [ 80, 49 ].
1013 - 1138 have Frattini factor [ 80, 50 ].
1139 - 1512 have Frattini factor [ 80, 51 ].
1513 - 1580 have Frattini factor [ 80, 52 ].
1581 - 1583 have Frattini factor [ 160, 234 ].
1584 - 1586 have Frattini factor [ 160, 235 ].
1587 - 1607 have Frattini factor [ 160, 236 ].
1608 - 1627 have Frattini factor [ 160, 237 ].
1628 - 1634 have Frattini factor [ 160, 238 ].
1635 - 1640 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.