# Groups of order 640

## Contents

See pages on algebraic structures of order 640| See pages on groups of a particular order

## Statistics at a glance

The number 640 has prime factorization $640 = 2^7 \cdot 5$. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 640 are solvable groups (specifically, finite solvable groups).

Quantity Value Explanation
number of groups up to isomorphism 21541
number of abelian groups up to isomorphism 15 (number of abelian groups of order $2^7$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 7) times (number of unordered integer partitions of 1) = $15 \times 1 = 15$. See also classification of finite abelian groups
number of nilpotent groups up to isomorphism 2328 (number of groups of order 128) times (number of groups of order 5) = $2328 \times 1 = 2328$. See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
number of solvable groups up to isomorphism 21541 There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 640 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 640 is part of GAP's SmallGroup library. Hence, any group of order 640 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 640 can be accessed as a list using GAP's AllSmallGroups function. However, the list size may be too large relative to the memory allocation given in typical GAP installations. To overcome this problem, use the IdsOfAllSmallGroups function which stores and manipulates only the group IDs, not the groups themselves.

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

gap> SmallGroupsInformation(640);

There are 21541 groups of order 640.
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 - 402 have Frattini factor [ 20, 4 ].
403 - 564 have Frattini factor [ 20, 5 ].
565 - 919 have Frattini factor [ 40, 12 ].
920 - 5021 have Frattini factor [ 40, 13 ].
5022 - 5854 have Frattini factor [ 40, 14 ].
5855 - 7108 have Frattini factor [ 80, 50 ].
7109 - 17941 have Frattini factor [ 80, 51 ].
17942 - 19094 have Frattini factor [ 80, 52 ].
19095 - 19099 have Frattini factor [ 160, 234 ].
19100 - 19104 have Frattini factor [ 160, 235 ].
19105 - 19649 have Frattini factor [ 160, 236 ].
19650 - 21284 have Frattini factor [ 160, 237 ].
21285 - 21453 have Frattini factor [ 160, 238 ].
21454 - 21456 have Frattini factor [ 320, 1635 ].
21457 - 21467 have Frattini factor [ 320, 1636 ].
21468 - 21474 have Frattini factor [ 320, 1637 ].
21475 - 21501 have Frattini factor [ 320, 1638 ].
21502 - 21526 have Frattini factor [ 320, 1639 ].
21527 - 21535 have Frattini factor [ 320, 1640 ].
21536 - 21541 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.