# Groups of order 864

## Contents

See pages on algebraic structures of order 864| See pages on groups of a particular order

## Statistics at a glance

The number 864 has the following prime factorization

$\! 864 = 2^5 \cdot 3^3 = 32 \cdot 27$

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 4725
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 21 (number of abelian groups of order $2^5$) times (number of abelian groups of order $3^3$) = (number of unordered integer partitions of 5) times (number of unordered integer partitions of 3) = $7 \times 3 = 21$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 255 (number of groups of order 32) times (number of groups of order 27) = $51 \times 5 = 255$. 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 (i.e., finite supersolvable groups) up to isomorphism 3581
Number of solvable groups (i.e., finite solvable groups) up to isomorphism 4725 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 864 is part of GAP's SmallGroup library. Hence, any group of order 864 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 864 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(864);

There are 4725 groups of order 864.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 - 44 have Frattini factor [ 12, 4 ].
45 - 63 have Frattini factor [ 12, 5 ].
64 - 67 have Frattini factor [ 18, 3 ].
68 - 69 have Frattini factor [ 18, 4 ].
70 - 72 have Frattini factor [ 18, 5 ].
73 - 76 have Frattini factor [ 24, 12 ].
77 - 83 have Frattini factor [ 24, 13 ].
84 - 169 have Frattini factor [ 24, 14 ].
170 - 193 have Frattini factor [ 24, 15 ].
194 has Frattini factor [ 36, 9 ].
195 - 302 have Frattini factor [ 36, 10 ].
303 - 307 have Frattini factor [ 36, 11 ].
308 - 471 have Frattini factor [ 36, 12 ].
472 - 553 have Frattini factor [ 36, 13 ].
554 - 610 have Frattini factor [ 36, 14 ].
611 - 621 have Frattini factor [ 48, 48 ].
622 - 628 have Frattini factor [ 48, 49 ].
629 - 630 have Frattini factor [ 48, 50 ].
631 - 645 have Frattini factor [ 48, 51 ].
646 - 651 have Frattini factor [ 48, 52 ].
652 has Frattini factor [ 54, 12 ].
653 has Frattini factor [ 54, 13 ].
654 has Frattini factor [ 54, 14 ].
655 has Frattini factor [ 54, 15 ].
656 has Frattini factor [ 72, 39 ].
657 - 674 have Frattini factor [ 72, 40 ].
675 - 679 have Frattini factor [ 72, 41 ].
680 - 695 have Frattini factor [ 72, 42 ].
696 - 707 have Frattini factor [ 72, 43 ].
708 - 736 have Frattini factor [ 72, 44 ].
737 - 761 have Frattini factor [ 72, 45 ].
762 - 1291 have Frattini factor [ 72, 46 ].
1292 - 1327 have Frattini factor [ 72, 47 ].
1328 - 1671 have Frattini factor [ 72, 48 ].
1672 - 1843 have Frattini factor [ 72, 49 ].
1844 - 1915 have Frattini factor [ 72, 50 ].
1916 has Frattini factor [ 96, 226 ].
1917 has Frattini factor [ 96, 227 ].
1918 has Frattini factor [ 96, 228 ].
1919 has Frattini factor [ 96, 229 ].
1920 has Frattini factor [ 96, 230 ].
1921 has Frattini factor [ 96, 231 ].
1922 has Frattini factor [ 108, 36 ].
1923 has Frattini factor [ 108, 37 ].
1924 - 1964 have Frattini factor [ 108, 38 ].
1965 - 2031 have Frattini factor [ 108, 39 ].
2032 - 2050 have Frattini factor [ 108, 40 ].
2051 has Frattini factor [ 108, 41 ].
2052 - 2092 have Frattini factor [ 108, 42 ].
2093 - 2133 have Frattini factor [ 108, 43 ].
2134 - 2174 have Frattini factor [ 108, 44 ].
2175 - 2193 have Frattini factor [ 108, 45 ].
2194 - 2196 have Frattini factor [ 144, 182 ].
2197 - 2241 have Frattini factor [ 144, 183 ].
2242 - 2253 have Frattini factor [ 144, 184 ].
2254 - 2260 have Frattini factor [ 144, 185 ].
2261 - 2283 have Frattini factor [ 144, 186 ].
2284 - 2288 have Frattini factor [ 144, 187 ].
2289 - 2332 have Frattini factor [ 144, 188 ].
2333 - 2365 have Frattini factor [ 144, 189 ].
2366 - 2409 have Frattini factor [ 144, 190 ].
2410 - 2422 have Frattini factor [ 144, 191 ].
2423 - 2512 have Frattini factor [ 144, 192 ].
2513 - 2547 have Frattini factor [ 144, 193 ].
2548 - 2557 have Frattini factor [ 144, 194 ].
2558 - 2617 have Frattini factor [ 144, 195 ].
2618 - 2647 have Frattini factor [ 144, 196 ].
2648 - 2665 have Frattini factor [ 144, 197 ].
2666 has Frattini factor [ 216, 153 ].
2667 has Frattini factor [ 216, 154 ].
2668 has Frattini factor [ 216, 155 ].
2669 - 2699 have Frattini factor [ 216, 156 ].
2700 - 2717 have Frattini factor [ 216, 157 ].
2718 - 2746 have Frattini factor [ 216, 158 ].
2747 - 2764 have Frattini factor [ 216, 159 ].
2765 - 2769 have Frattini factor [ 216, 160 ].
2770 - 2778 have Frattini factor [ 216, 161 ].
2779 - 3004 have Frattini factor [ 216, 162 ].
3005 - 3008 have Frattini factor [ 216, 163 ].
3009 - 3012 have Frattini factor [ 216, 164 ].
3013 - 3016 have Frattini factor [ 216, 165 ].
3017 - 3023 have Frattini factor [ 216, 166 ].
3024 - 3030 have Frattini factor [ 216, 167 ].
3031 - 3055 have Frattini factor [ 216, 168 ].
3056 - 3080 have Frattini factor [ 216, 169 ].
3081 - 3275 have Frattini factor [ 216, 170 ].
3276 - 3610 have Frattini factor [ 216, 171 ].
3611 - 3696 have Frattini factor [ 216, 172 ].
3697 - 3703 have Frattini factor [ 216, 173 ].
3704 - 3789 have Frattini factor [ 216, 174 ].
3790 - 3875 have Frattini factor [ 216, 175 ].
3876 - 3961 have Frattini factor [ 216, 176 ].
3962 - 3985 have Frattini factor [ 216, 177 ].
3986 - 3989 have Frattini factor [ 288, 1024 ].
3990 - 3993 have Frattini factor [ 288, 1025 ].
3994 - 3996 have Frattini factor [ 288, 1026 ].
3997 has Frattini factor [ 288, 1027 ].
3998 - 4000 have Frattini factor [ 288, 1028 ].
4001 - 4005 have Frattini factor [ 288, 1029 ].
4006 has Frattini factor [ 288, 1030 ].
4007 has Frattini factor [ 288, 1031 ].
4008 has Frattini factor [ 288, 1032 ].
4009 - 4012 have Frattini factor [ 288, 1033 ].
4013 - 4015 have Frattini factor [ 288, 1034 ].
4016 - 4019 have Frattini factor [ 288, 1035 ].
4020 - 4022 have Frattini factor [ 288, 1036 ].
4023 - 4026 have Frattini factor [ 288, 1037 ].
4027 - 4030 have Frattini factor [ 288, 1038 ].
4031 has Frattini factor [ 288, 1039 ].
4032 - 4033 have Frattini factor [ 288, 1040 ].
4034 - 4038 have Frattini factor [ 288, 1041 ].
4039 - 4043 have Frattini factor [ 288, 1042 ].
4044 - 4047 have Frattini factor [ 288, 1043 ].
4048 - 4049 have Frattini factor [ 288, 1044 ].
4050 - 4052 have Frattini factor [ 288, 1045 ].
4053 has Frattini factor [ 432, 734 ].
4054 has Frattini factor [ 432, 735 ].
4055 - 4061 have Frattini factor [ 432, 736 ].
4062 - 4064 have Frattini factor [ 432, 737 ].
4065 - 4067 have Frattini factor [ 432, 738 ].
4068 - 4070 have Frattini factor [ 432, 739 ].
4071 - 4073 have Frattini factor [ 432, 740 ].
4074 - 4112 have Frattini factor [ 432, 741 ].
4113 - 4121 have Frattini factor [ 432, 742 ].
4122 - 4124 have Frattini factor [ 432, 743 ].
4125 - 4127 have Frattini factor [ 432, 744 ].
4128 - 4142 have Frattini factor [ 432, 745 ].
4143 - 4157 have Frattini factor [ 432, 746 ].
4158 - 4172 have Frattini factor [ 432, 747 ].
4173 - 4183 have Frattini factor [ 432, 748 ].
4184 - 4194 have Frattini factor [ 432, 749 ].
4195 - 4196 have Frattini factor [ 432, 750 ].
4197 - 4203 have Frattini factor [ 432, 751 ].
4204 - 4210 have Frattini factor [ 432, 752 ].
4211 - 4241 have Frattini factor [ 432, 753 ].
4242 - 4264 have Frattini factor [ 432, 754 ].
4265 - 4299 have Frattini factor [ 432, 755 ].
4300 - 4322 have Frattini factor [ 432, 756 ].
4323 - 4327 have Frattini factor [ 432, 757 ].
4328 - 4336 have Frattini factor [ 432, 758 ].
4337 - 4407 have Frattini factor [ 432, 759 ].
4408 - 4418 have Frattini factor [ 432, 760 ].
4419 - 4429 have Frattini factor [ 432, 761 ].
4430 - 4440 have Frattini factor [ 432, 762 ].
4441 - 4451 have Frattini factor [ 432, 763 ].
4452 - 4462 have Frattini factor [ 432, 764 ].
4463 - 4475 have Frattini factor [ 432, 765 ].
4476 - 4488 have Frattini factor [ 432, 766 ].
4489 - 4525 have Frattini factor [ 432, 767 ].
4526 - 4578 have Frattini factor [ 432, 768 ].
4579 - 4600 have Frattini factor [ 432, 769 ].
4601 - 4607 have Frattini factor [ 432, 770 ].
4608 - 4609 have Frattini factor [ 432, 771 ].
4610 - 4624 have Frattini factor [ 432, 772 ].
4625 - 4639 have Frattini factor [ 432, 773 ].
4640 - 4654 have Frattini factor [ 432, 774 ].
4655 - 4660 have Frattini factor [ 432, 775 ].
4661 - 4725 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.