# Groups of order 1000

## Contents

See pages on algebraic structures of order 1000| See pages on groups of a particular order

## Statistics at a glance

The number 1000 has prime factors 2 and 5. The prime factorization is: $\! 1000 = 2^3 \cdot 5^3 = 8 \cdot 125$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity Value Explanation
Total number of groups up to isomorphism 199
Number of abelian groups up to isomorphism 9 (number of groups of order $2^3$) $\times$ (number of groups of order $5^3$) = $3 \times 3 = 9$.
See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 25 (number of groups of order 8) $\times$ (number of groups of order 125 -- see also groups of prime-cube order) = $5 \times 5 = 25$.
See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
Number of supersolvable groups up to isomorphism 187
Number of solvable groups up to isomorphism 199 There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups up to isomorphism 0

## GAP implementation

The order 1000 is part of GAP's SmallGroup library. Hence, any group of order 1000 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 1000 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(1000);

There are 199 groups of order 1000.
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 - 8 have Frattini factor [ 20, 4 ].
9 - 11 have Frattini factor [ 20, 5 ].
12 has Frattini factor [ 40, 12 ].
13 has Frattini factor [ 40, 13 ].
14 has Frattini factor [ 40, 14 ].
15 - 18 have Frattini factor [ 50, 3 ].
19 - 20 have Frattini factor [ 50, 4 ].
21 - 23 have Frattini factor [ 50, 5 ].
24 - 27 have Frattini factor [ 100, 9 ].
28 - 30 have Frattini factor [ 100, 10 ].
31 - 32 have Frattini factor [ 100, 11 ].
33 - 34 have Frattini factor [ 100, 12 ].
35 - 46 have Frattini factor [ 100, 13 ].
47 - 66 have Frattini factor [ 100, 14 ].
67 - 76 have Frattini factor [ 100, 15 ].
77 - 85 have Frattini factor [ 100, 16 ].
86 has Frattini factor [ 200, 40 ].
87 - 89 have Frattini factor [ 200, 41 ].
90 - 91 have Frattini factor [ 200, 42 ].
92 has Frattini factor [ 200, 43 ].
93 has Frattini factor [ 200, 44 ].
94 - 97 have Frattini factor [ 200, 45 ].
98 - 100 have Frattini factor [ 200, 46 ].
101 - 102 have Frattini factor [ 200, 47 ].
103 - 104 have Frattini factor [ 200, 48 ].
105 - 106 have Frattini factor [ 200, 49 ].
107 - 110 have Frattini factor [ 200, 50 ].
111 - 112 have Frattini factor [ 200, 51 ].
113 - 115 have Frattini factor [ 200, 52 ].
116 has Frattini factor [ 250, 12 ].
117 has Frattini factor [ 250, 13 ].
118 has Frattini factor [ 250, 14 ].
119 has Frattini factor [ 250, 15 ].
120 has Frattini factor [ 500, 41 ].
121 has Frattini factor [ 500, 42 ].
122 has Frattini factor [ 500, 43 ].
123 has Frattini factor [ 500, 44 ].
124 has Frattini factor [ 500, 45 ].
125 has Frattini factor [ 500, 46 ].
126 has Frattini factor [ 500, 47 ].
127 has Frattini factor [ 500, 48 ].
128 has Frattini factor [ 500, 49 ].
129 - 133 have Frattini factor [ 500, 50 ].
134 - 140 have Frattini factor [ 500, 51 ].
141 - 143 have Frattini factor [ 500, 52 ].
144 - 148 have Frattini factor [ 500, 53 ].
149 - 153 have Frattini factor [ 500, 54 ].
154 - 158 have Frattini factor [ 500, 55 ].
159 - 161 have Frattini factor [ 500, 56 ].
162 - 199 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.