Groups of order 80

Statistics at a glance
The number 80 has prime factorization $$80 = 2^4 \cdot 5$$. There are only two prime factors, and order has only two prime factors implies solvable, so all groups of order 80 are solvable groups (specifically, finite solvable groups).

GAP implementation
The order 80 is part of GAP's SmallGroup library. Hence, all groups of order 80 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(80);

There are 52 groups of order 80. 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 - 19 have Frattini factor [ 20, 4 ]. 20 - 27 have Frattini factor [ 20, 5 ]. 28 - 34 have Frattini factor [ 40, 12 ]. 35 - 44 have Frattini factor [ 40, 13 ]. 45 - 48 have Frattini factor [ 40, 14 ]. 49 - 52 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.