# Groups of order 1280

## Contents

See pages on algebraic structures of order 1280| See pages on groups of a particular order

## Statistics at a glance

The number 80 has prime factorization $1280 = 2^8 \cdot 5$. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 1280 are solvable groups (specifically, finite solvable groups).

Quantity Value Explanation
number of groups up to isomorphism 1116461
number of abelian groups up to isomorphism 22 (number of abelian groups of order $2^8$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 8) times (number of unordered integer partitions of 1) = $22 \times 1 = 22$. See also classification of finite abelian groups
number of nilpotent groups up to isomorphism 56092 (number of groups of order 256) times (number of groups of order 5) = $56092 \times 1 = 56092$
number of solvable groups up to isomorphism 1116461 There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 1280 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 1280 is part of GAP's SmallGroup library. Hence, all groups of order 1280 can be constructed using the SmallGroup function and have group IDs. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

gap> SmallGroupsInformation(1280);

There are 1116461 groups of order 1280.
They are sorted by normal Sylow subgroups.
1 - 56092 are the nilpotent groups.
56093 - 1083472 have a normal Sylow 5-subgroup
with centralizer of index 2.
1083473 - 1116308 have a normal Sylow 5-subgroup
with centralizer of index 4.
1116309 - 1116357 have a normal Sylow 2-subgroup.
1116358 - 1116461 have no normal Sylow subgroup.

This size belongs to layer 3 of the SmallGroups library.
IdSmallGroup is available for this size.