# Groups of order 768

## Contents

See pages on algebraic structures of order 768| See pages on groups of a particular order

## Statistics at a glance

The number 768 has prime factorization $768 = 2^8 \cdot 3$. Since there are only two prime factors, and order has only two prime factors implies solvable, all groups of this order are solvable groups.

Quantity Value Explanation
Number of groups up to isomorphism 1090235
Number of abelian groups up to isomorphism 22 Equals (number of abelian groups of order $2^8$) times (number of abelian groups of order 3) = (number of unordered integer partitions of $8$) times (number of unordered integer partitions of 1) = $22 \times 1 = 22$
Number of nilpotent groups up to isomorphism 56092 Equals (number of groups of order 256) times (number of groups of order 3) = $56092 \times 1 = 56092$
Number of solvable groups up to isomorphism 1090235 All groups of this order are solvable, because order has only two prime factors implies solvable
Number of simple groups up to isomorphism 0

## GAP implementation

The order 768 is part of GAP's SmallGroup library. Hence, all groups of order 768 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.

However, note that with the memory allocations of most computers on which GAP is run, GAP will not allow storing all the groups or their IDs in a list, since the list is too long to store and process as a list.

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

gap> SmallGroupsInformation(768);

There are 1090235 groups of order 768.
They are sorted by normal Sylow subgroups.
1 - 56092 are the nilpotent groups.
56093 - 1083472 have a normal Sylow 3-subgroup
with centralizer of index 2.
1083473 - 1085323 have a normal Sylow 2-subgroup.
1085324 - 1090235 have no normal Sylow subgroup.

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