# Groups of order 2000

## Contents

See pages on algebraic structures of order 2000| See pages on groups of a particular order

## Statistics at a glance

The number 2000 has prime factors 2 and 5. The prime factorization is as follows: $\! 2000 = 2^4 \cdot 5^3 = 16 \cdot 125$

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 963
Number of abelian groups up to isomorphism 15 (number of abelian groups of order $2^4$) $\times$ (number of abelian groups of order $5^3$) = (number of unordered integer partitions of 4) $\times$ (number of unordered integer partitions of 3) = $5 \times 3 = 15$.
See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 70 (number of groups of order 16) $\times$ (number of groups of order 125) = $14 \times 5 = 70$.
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 894
Number of solvable groups up to isomorphism 963 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.

## GAP implementation

The order 2000 is part of GAP's SmallGroup library. Hence, any group of order 2000 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 2000 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(2000);

There are 963 groups of order 2000.
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 Frattini factor [ 50, 3 ].
53 - 54 have Frattini factor [ 50, 4 ].
55 - 57 have Frattini factor [ 50, 5 ].
58 has Frattini factor [ 80, 49 ].
59 has Frattini factor [ 80, 50 ].
60 has Frattini factor [ 80, 51 ].
61 has Frattini factor [ 80, 52 ].
62 - 65 have Frattini factor [ 100, 9 ].
66 - 68 have Frattini factor [ 100, 10 ].
69 - 70 have Frattini factor [ 100, 11 ].
71 - 72 have Frattini factor [ 100, 12 ].
73 - 113 have Frattini factor [ 100, 13 ].
114 - 177 have Frattini factor [ 100, 14 ].
178 - 209 have Frattini factor [ 100, 15 ].
210 - 233 have Frattini factor [ 100, 16 ].
234 has Frattini factor [ 200, 40 ].
235 - 255 have Frattini factor [ 200, 41 ].
256 - 267 have Frattini factor [ 200, 42 ].
268 - 272 have Frattini factor [ 200, 43 ].
273 has Frattini factor [ 200, 44 ].
274 - 301 have Frattini factor [ 200, 45 ].
302 - 322 have Frattini factor [ 200, 46 ].
323 - 336 have Frattini factor [ 200, 47 ].
337 - 350 have Frattini factor [ 200, 48 ].
351 - 395 have Frattini factor [ 200, 49 ].
396 - 435 have Frattini factor [ 200, 50 ].
436 - 455 have Frattini factor [ 200, 51 ].
456 - 467 have Frattini factor [ 200, 52 ].
468 has Frattini factor [ 250, 12 ].
469 has Frattini factor [ 250, 13 ].
470 has Frattini factor [ 250, 14 ].
471 has Frattini factor [ 250, 15 ].
472 - 473 have Frattini factor [ 400, 205 ].
474 has Frattini factor [ 400, 206 ].
475 has Frattini factor [ 400, 207 ].
476 has Frattini factor [ 400, 208 ].
477 - 479 have Frattini factor [ 400, 209 ].
480 - 481 have Frattini factor [ 400, 210 ].
482 has Frattini factor [ 400, 211 ].
483 has Frattini factor [ 400, 212 ].
484 - 488 have Frattini factor [ 400, 213 ].
489 - 492 have Frattini factor [ 400, 214 ].
493 - 495 have Frattini factor [ 400, 215 ].
496 - 497 have Frattini factor [ 400, 216 ].
498 - 499 have Frattini factor [ 400, 217 ].
500 - 501 have Frattini factor [ 400, 218 ].
502 - 505 have Frattini factor [ 400, 219 ].
506 - 507 have Frattini factor [ 400, 220 ].
508 - 510 have Frattini factor [ 400, 221 ].
511 has Frattini factor [ 500, 41 ].
512 has Frattini factor [ 500, 42 ].
513 has Frattini factor [ 500, 43 ].
514 has Frattini factor [ 500, 44 ].
515 has Frattini factor [ 500, 45 ].
516 has Frattini factor [ 500, 46 ].
517 has Frattini factor [ 500, 47 ].
518 has Frattini factor [ 500, 48 ].
519 has Frattini factor [ 500, 49 ].
520 - 535 have Frattini factor [ 500, 50 ].
536 - 560 have Frattini factor [ 500, 51 ].
561 - 568 have Frattini factor [ 500, 52 ].
569 - 584 have Frattini factor [ 500, 53 ].
585 - 600 have Frattini factor [ 500, 54 ].
601 - 616 have Frattini factor [ 500, 55 ].
617 - 624 have Frattini factor [ 500, 56 ].
625 has Frattini factor [ 1000, 162 ].
626 has Frattini factor [ 1000, 163 ].
627 has Frattini factor [ 1000, 164 ].
628 has Frattini factor [ 1000, 165 ].
629 - 633 have Frattini factor [ 1000, 166 ].
634 - 640 have Frattini factor [ 1000, 167 ].
641 - 647 have Frattini factor [ 1000, 168 ].
648 - 654 have Frattini factor [ 1000, 169 ].
655 - 661 have Frattini factor [ 1000, 170 ].
662 - 668 have Frattini factor [ 1000, 171 ].
669 - 673 have Frattini factor [ 1000, 172 ].
674 - 678 have Frattini factor [ 1000, 173 ].
679 - 683 have Frattini factor [ 1000, 174 ].
684 - 690 have Frattini factor [ 1000, 175 ].
691 - 697 have Frattini factor [ 1000, 176 ].
698 - 704 have Frattini factor [ 1000, 177 ].
705 - 709 have Frattini factor [ 1000, 178 ].
710 - 714 have Frattini factor [ 1000, 179 ].
715 - 721 have Frattini factor [ 1000, 180 ].
722 has Frattini factor [ 1000, 181 ].
723 - 724 have Frattini factor [ 1000, 182 ].
725 - 743 have Frattini factor [ 1000, 183 ].
744 - 750 have Frattini factor [ 1000, 184 ].
751 - 757 have Frattini factor [ 1000, 185 ].
758 - 764 have Frattini factor [ 1000, 186 ].
765 - 771 have Frattini factor [ 1000, 187 ].
772 - 778 have Frattini factor [ 1000, 188 ].
779 - 785 have Frattini factor [ 1000, 189 ].
786 - 792 have Frattini factor [ 1000, 190 ].
793 - 799 have Frattini factor [ 1000, 191 ].
800 - 806 have Frattini factor [ 1000, 192 ].
807 - 824 have Frattini factor [ 1000, 193 ].
825 - 851 have Frattini factor [ 1000, 194 ].
852 - 861 have Frattini factor [ 1000, 195 ].
862 - 871 have Frattini factor [ 1000, 196 ].
872 - 881 have Frattini factor [ 1000, 197 ].
882 - 891 have Frattini factor [ 1000, 198 ].
892 - 895 have Frattini factor [ 1000, 199 ].
896 - 963 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.