Groups of order 768

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.

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.