# Groups of order 1512

## Contents

See pages on algebraic structures of order 1512| See pages on groups of a particular order

## Statistics at a glance

The number 1512 has prime factors 2, 3, and 7. The prime factorization is as follows:

$\! 1512 = 2^3 \cdot 3^3 \cdot 7^1 = 8 \cdot 27 \cdot 7$

There are both solvable and non-solvable groups of this order.

For a non-solvable group, there are two possibilities for the collection of composition factors. One possibility is that the non-abelian composition factor is projective special linear group:PSL(3,2) (order 168), and there are two copies of cyclic group:Z3. The other is that the non-abelian composition factor is projective special linear group:PSL(2,8) (order 504), and there is one copy of cyclic group:Z3.

Quantity Value Explanation
Total number of groups up to isomorphism 889
Number of abelian groups up to isomorphism 9 (number of abelian groups of order $2^3$) $\times$ (number of abelian groups of order $3^3$) $\times$ (number of abelian groups of order $7^1$) = (number of unordered integer partitions of 3) $\times$ (number of unordered integer partitions of 3) $\times$ (number of unordered integer partitions of 1) = $3 \times 3 \times 1 = 9$.
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 27) $\times$ (number of groups of order 7) = $5 \times 5 \times 1 = 25$.
Number of supersolvable groups up to isomorphism 674
Number of solvable groups up to isomorphism 885 (see note on non-solvable groups)
Number of non-solvable groups up to isomorphism 4 The only non-solvable groups are the groups with either projective special linear group:PSL(3,2) or projective special linear group:PSL(2,8) as a simple non-abelian composition factor. There are two groups of each type.
For PSL(3,2): direct product of PSL(3,2) and Z9, direct product of PSL(3,2) and E9
For PSL(2,8): direct product of PSL(2,8) and Z3, Ree group:Ree(3) (which is isomorphic to the projective semilinear group $P\Gamma L(2,8)$)

## GAP implementation

The order 1512 is part of GAP's SmallGroup library. Hence, any group of order 1512 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 1512 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(1512);

There are 889 groups of order 1512.
They are sorted by their Frattini factors.
1 has Frattini factor [ 42, 1 ].
2 has Frattini factor [ 42, 2 ].
3 has Frattini factor [ 42, 3 ].
4 has Frattini factor [ 42, 4 ].
5 has Frattini factor [ 42, 5 ].
6 has Frattini factor [ 42, 6 ].
7 - 11 have Frattini factor [ 84, 7 ].
12 - 18 have Frattini factor [ 84, 8 ].
19 - 21 have Frattini factor [ 84, 9 ].
22 has Frattini factor [ 84, 10 ].
23 has Frattini factor [ 84, 11 ].
24 - 28 have Frattini factor [ 84, 12 ].
29 - 33 have Frattini factor [ 84, 13 ].
34 - 38 have Frattini factor [ 84, 14 ].
39 - 41 have Frattini factor [ 84, 15 ].
42 - 47 have Frattini factor [ 126, 7 ].
48 - 52 have Frattini factor [ 126, 8 ].
53 - 57 have Frattini factor [ 126, 9 ].
58 - 63 have Frattini factor [ 126, 10 ].
64 - 66 have Frattini factor [ 126, 11 ].
67 - 70 have Frattini factor [ 126, 12 ].
71 - 74 have Frattini factor [ 126, 13 ].
75 - 76 have Frattini factor [ 126, 14 ].
77 - 78 have Frattini factor [ 126, 15 ].
79 - 81 have Frattini factor [ 126, 16 ].
82 has Frattini factor [ 168, 43 ].
83 has Frattini factor [ 168, 44 ].
84 has Frattini factor [ 168, 45 ].
85 has Frattini factor [ 168, 46 ].
86 has Frattini factor [ 168, 47 ].
87 has Frattini factor [ 168, 48 ].
88 has Frattini factor [ 168, 49 ].
89 has Frattini factor [ 168, 50 ].
90 has Frattini factor [ 168, 51 ].
91 has Frattini factor [ 168, 52 ].
92 has Frattini factor [ 168, 53 ].
93 has Frattini factor [ 168, 54 ].
94 has Frattini factor [ 168, 55 ].
95 has Frattini factor [ 168, 56 ].
96 has Frattini factor [ 168, 57 ].
97 - 131 have Frattini factor [ 252, 26 ].
132 - 140 have Frattini factor [ 252, 27 ].
141 - 170 have Frattini factor [ 252, 28 ].
171 - 195 have Frattini factor [ 252, 29 ].
196 - 220 have Frattini factor [ 252, 30 ].
221 has Frattini factor [ 252, 31 ].
222 has Frattini factor [ 252, 32 ].
223 - 250 have Frattini factor [ 252, 33 ].
251 - 264 have Frattini factor [ 252, 34 ].
265 - 276 have Frattini factor [ 252, 35 ].
277 - 297 have Frattini factor [ 252, 36 ].
298 - 309 have Frattini factor [ 252, 37 ].
310 - 327 have Frattini factor [ 252, 38 ].
328 - 332 have Frattini factor [ 252, 39 ].
333 - 338 have Frattini factor [ 252, 40 ].
339 - 353 have Frattini factor [ 252, 41 ].
354 - 373 have Frattini factor [ 252, 42 ].
374 - 393 have Frattini factor [ 252, 43 ].
394 - 403 have Frattini factor [ 252, 44 ].
404 - 413 have Frattini factor [ 252, 45 ].
414 - 422 have Frattini factor [ 252, 46 ].
423 has Frattini factor [ 378, 47 ].
424 has Frattini factor [ 378, 48 ].
425 has Frattini factor [ 378, 49 ].
426 has Frattini factor [ 378, 50 ].
427 has Frattini factor [ 378, 51 ].
428 has Frattini factor [ 378, 52 ].
429 has Frattini factor [ 378, 53 ].
430 has Frattini factor [ 378, 54 ].
431 has Frattini factor [ 378, 55 ].
432 has Frattini factor [ 378, 56 ].
433 has Frattini factor [ 378, 57 ].
434 has Frattini factor [ 378, 58 ].
435 has Frattini factor [ 378, 59 ].
436 has Frattini factor [ 378, 60 ].
437 has Frattini factor [ 504, 157 ].
438 - 443 have Frattini factor [ 504, 158 ].
444 - 448 have Frattini factor [ 504, 159 ].
449 - 453 have Frattini factor [ 504, 160 ].
454 - 462 have Frattini factor [ 504, 161 ].
463 - 467 have Frattini factor [ 504, 162 ].
468 - 476 have Frattini factor [ 504, 163 ].
477 has Frattini factor [ 504, 164 ].
478 has Frattini factor [ 504, 165 ].
479 has Frattini factor [ 504, 166 ].
480 has Frattini factor [ 504, 167 ].
481 has Frattini factor [ 504, 168 ].
482 has Frattini factor [ 504, 169 ].
483 has Frattini factor [ 504, 170 ].
484 has Frattini factor [ 504, 171 ].
485 - 486 have Frattini factor [ 504, 172 ].
487 - 489 have Frattini factor [ 504, 173 ].
490 - 493 have Frattini factor [ 504, 174 ].
494 - 497 have Frattini factor [ 504, 175 ].
498 - 500 have Frattini factor [ 504, 176 ].
501 - 503 have Frattini factor [ 504, 177 ].
504 - 509 have Frattini factor [ 504, 178 ].
510 - 514 have Frattini factor [ 504, 179 ].
515 - 519 have Frattini factor [ 504, 180 ].
520 - 524 have Frattini factor [ 504, 181 ].
525 - 530 have Frattini factor [ 504, 182 ].
531 - 534 have Frattini factor [ 504, 183 ].
535 - 538 have Frattini factor [ 504, 184 ].
539 - 543 have Frattini factor [ 504, 185 ].
544 - 548 have Frattini factor [ 504, 186 ].
549 has Frattini factor [ 504, 187 ].
550 has Frattini factor [ 504, 188 ].
551 - 554 have Frattini factor [ 504, 189 ].
555 - 556 have Frattini factor [ 504, 190 ].
557 - 558 have Frattini factor [ 504, 191 ].
559 - 561 have Frattini factor [ 504, 192 ].
562 - 563 have Frattini factor [ 504, 193 ].
564 - 569 have Frattini factor [ 504, 194 ].
570 - 574 have Frattini factor [ 504, 195 ].
575 - 580 have Frattini factor [ 504, 196 ].
581 - 583 have Frattini factor [ 504, 197 ].
584 - 587 have Frattini factor [ 504, 198 ].
588 - 591 have Frattini factor [ 504, 199 ].
592 - 593 have Frattini factor [ 504, 200 ].
594 - 595 have Frattini factor [ 504, 201 ].
596 - 598 have Frattini factor [ 504, 202 ].
599 has Frattini factor [ 756, 150 ].
600 has Frattini factor [ 756, 151 ].
601 - 607 have Frattini factor [ 756, 152 ].
608 - 614 have Frattini factor [ 756, 153 ].
615 - 619 have Frattini factor [ 756, 154 ].
620 - 626 have Frattini factor [ 756, 155 ].
627 - 631 have Frattini factor [ 756, 156 ].
632 has Frattini factor [ 756, 157 ].
633 - 637 have Frattini factor [ 756, 158 ].
638 - 642 have Frattini factor [ 756, 159 ].
643 - 647 have Frattini factor [ 756, 160 ].
648 - 652 have Frattini factor [ 756, 161 ].
653 - 657 have Frattini factor [ 756, 162 ].
658 has Frattini factor [ 756, 163 ].
659 has Frattini factor [ 756, 164 ].
660 has Frattini factor [ 756, 165 ].
661 has Frattini factor [ 756, 166 ].
662 - 668 have Frattini factor [ 756, 167 ].
669 - 675 have Frattini factor [ 756, 168 ].
676 - 680 have Frattini factor [ 756, 169 ].
681 - 687 have Frattini factor [ 756, 170 ].
688 - 692 have Frattini factor [ 756, 171 ].
693 - 699 have Frattini factor [ 756, 172 ].
700 - 706 have Frattini factor [ 756, 173 ].
707 - 713 have Frattini factor [ 756, 174 ].
714 - 720 have Frattini factor [ 756, 175 ].
721 - 727 have Frattini factor [ 756, 176 ].
728 - 730 have Frattini factor [ 756, 177 ].
731 - 735 have Frattini factor [ 756, 178 ].
736 - 738 have Frattini factor [ 756, 179 ].
739 has Frattini factor [ 756, 180 ].
740 has Frattini factor [ 756, 181 ].
741 - 745 have Frattini factor [ 756, 182 ].
746 - 750 have Frattini factor [ 756, 183 ].
751 - 755 have Frattini factor [ 756, 184 ].
756 - 760 have Frattini factor [ 756, 185 ].
761 - 765 have Frattini factor [ 756, 186 ].
766 - 770 have Frattini factor [ 756, 187 ].
771 - 775 have Frattini factor [ 756, 188 ].
776 - 778 have Frattini factor [ 756, 189 ].
779 - 889 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.