# Groups of order 1056

## Contents

See pages on algebraic structures of order 1056| See pages on groups of a particular order

## Statistics at a glance

The number 1056 has prime factors 2,3,11 with prime factorization: $\! 1056 = 2^5 \cdot 3^1 \cdot 11^1 = 32 \cdot 3 \cdot 11$

Quantity Value Explanation
Total number of groups up to isomorphism 1028
Number of abelian groups up to isomorphism 7 (number of abelian groups of order $2^5$) $\times$ (number of abelian groups of order $3^1$) $\times$ (number of abelian groups of order $11^1$) = (number of unordered integer partitions of 5) $\times$ (number of unordered integer partitions of 1) $\times$ (number of unordered integer partitions of 1) = $7 \times 1 \times 1 = 7$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 51 (number of groups of order 32) $\times$ (number of groups of order 3) $\times$ (number of groups of order 11) = $51 \times 1 \times 1 = 51$.
Number of supersolvable groups up to isomorphism 939
Number of solvable groups up to isomorphism 1028 all groups of this order are solvable.
Number of simple non-abelian groups up to isomorphism 0

## GAP implementation

The order 1056 is part of GAP's SmallGroup library. Hence, any group of order 1056 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 1056 can be accessed as a list using GAP's AllSmallGroups function.

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

gap> SmallGroupsInformation(1056);

There are 1028 groups of order 1056.
They are sorted by their Frattini factors.
1 has Frattini factor [ 66, 1 ].
2 has Frattini factor [ 66, 2 ].
3 has Frattini factor [ 66, 3 ].
4 has Frattini factor [ 66, 4 ].
5 - 71 have Frattini factor [ 132, 5 ].
72 has Frattini factor [ 132, 6 ].
73 - 113 have Frattini factor [ 132, 7 ].
114 - 154 have Frattini factor [ 132, 8 ].
155 - 195 have Frattini factor [ 132, 9 ].
196 - 214 have Frattini factor [ 132, 10 ].
215 - 218 have Frattini factor [ 264, 31 ].
219 - 222 have Frattini factor [ 264, 32 ].
223 - 229 have Frattini factor [ 264, 33 ].
230 - 564 have Frattini factor [ 264, 34 ].
565 - 571 have Frattini factor [ 264, 35 ].
572 - 657 have Frattini factor [ 264, 36 ].
658 - 743 have Frattini factor [ 264, 37 ].
744 - 829 have Frattini factor [ 264, 38 ].
830 - 853 have Frattini factor [ 264, 39 ].
854 - 868 have Frattini factor [ 528, 160 ].
869 - 879 have Frattini factor [ 528, 161 ].
880 - 890 have Frattini factor [ 528, 162 ].
891 - 901 have Frattini factor [ 528, 163 ].
902 - 954 have Frattini factor [ 528, 164 ].
955 - 961 have Frattini factor [ 528, 165 ].
962 - 963 have Frattini factor [ 528, 166 ].
964 - 978 have Frattini factor [ 528, 167 ].
979 - 993 have Frattini factor [ 528, 168 ].
994 - 1008 have Frattini factor [ 528, 169 ].
1009 - 1014 have Frattini factor [ 528, 170 ].
1015 - 1028 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 5 of the SmallGroups library.
IdSmallGroup is available for this size.