# Groups of order 576

## Contents

This article gives information about, and links to more details on, groups of order 576
See pages on algebraic structures of order 576| See pages on groups of a particular order

## Statistics at a glance

The number 576 has the following prime factorization:

$\! 576 = 2^6 \cdot 3^2 = 64 \cdot 9$

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 8681
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 22 (number of abelian groups of order $2^6$) times (number of abelian groups of order $3^2$) = (number of unordered integer partitions of 6) times (number of unordered integer partitions of 2) = $11 \times 2 = 22$. 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 534 (number of groups of order 64) times (number of groups of order 9) = $267 \times 2 = 534$. 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 solvable groups (i.e., finite solvable groups) up to isomorphism 8681 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 0 Follows from all groups of the order being solvable.

## GAP implementation

The order 576 is part of GAP's SmallGroup library. Hence, any group of order 576 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 576 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(576);

There are 8681 groups of order 576.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 - 4 have Frattini factor [ 12, 3 ].
5 - 126 have Frattini factor [ 12, 4 ].
127 - 179 have Frattini factor [ 12, 5 ].
180 has Frattini factor [ 18, 3 ].
181 has Frattini factor [ 18, 4 ].
182 has Frattini factor [ 18, 5 ].
183 - 190 have Frattini factor [ 24, 12 ].
191 - 207 have Frattini factor [ 24, 13 ].
208 - 809 have Frattini factor [ 24, 14 ].
810 - 946 have Frattini factor [ 24, 15 ].
947 has Frattini factor [ 36, 9 ].
948 - 1069 have Frattini factor [ 36, 10 ].
1070 - 1071 have Frattini factor [ 36, 11 ].
1072 - 1193 have Frattini factor [ 36, 12 ].
1194 - 1315 have Frattini factor [ 36, 13 ].
1316 - 1368 have Frattini factor [ 36, 14 ].
1369 - 1416 have Frattini factor [ 48, 48 ].
1417 - 1444 have Frattini factor [ 48, 49 ].
1445 - 1450 have Frattini factor [ 48, 50 ].
1451 - 1824 have Frattini factor [ 48, 51 ].
1825 - 1892 have Frattini factor [ 48, 52 ].
1893 has Frattini factor [ 72, 39 ].
1894 - 1963 have Frattini factor [ 72, 40 ].
1964 - 1980 have Frattini factor [ 72, 41 ].
1981 - 1988 have Frattini factor [ 72, 42 ].
1989 - 1996 have Frattini factor [ 72, 43 ].
1997 - 2013 have Frattini factor [ 72, 44 ].
2014 - 2105 have Frattini factor [ 72, 45 ].
2106 - 3607 have Frattini factor [ 72, 46 ].
3608 - 3624 have Frattini factor [ 72, 47 ].
3625 - 4226 have Frattini factor [ 72, 48 ].
4227 - 4828 have Frattini factor [ 72, 49 ].
4829 - 4965 have Frattini factor [ 72, 50 ].
4966 - 4986 have Frattini factor [ 96, 226 ].
4987 - 4993 have Frattini factor [ 96, 227 ].
4994 - 5002 have Frattini factor [ 96, 228 ].
5003 - 5007 have Frattini factor [ 96, 229 ].
5008 - 5027 have Frattini factor [ 96, 230 ].
5028 - 5034 have Frattini factor [ 96, 231 ].
5035 - 5052 have Frattini factor [ 144, 182 ].
5053 - 5126 have Frattini factor [ 144, 183 ].
5127 - 5129 have Frattini factor [ 144, 184 ].
5130 - 5156 have Frattini factor [ 144, 185 ].
5157 - 5412 have Frattini factor [ 144, 186 ].
5413 - 5452 have Frattini factor [ 144, 187 ].
5453 - 5500 have Frattini factor [ 144, 188 ].
5501 - 5548 have Frattini factor [ 144, 189 ].
5549 - 5603 have Frattini factor [ 144, 190 ].
5604 - 5729 have Frattini factor [ 144, 191 ].
5730 - 7411 have Frattini factor [ 144, 192 ].
7412 - 7439 have Frattini factor [ 144, 193 ].
7440 - 7445 have Frattini factor [ 144, 194 ].
7446 - 7819 have Frattini factor [ 144, 195 ].
7820 - 8193 have Frattini factor [ 144, 196 ].
8194 - 8261 have Frattini factor [ 144, 197 ].
8262 has Frattini factor [ 192, 1537 ].
8263 has Frattini factor [ 192, 1538 ].
8264 has Frattini factor [ 192, 1539 ].
8265 has Frattini factor [ 192, 1540 ].
8266 has Frattini factor [ 192, 1541 ].
8267 has Frattini factor [ 192, 1542 ].
8268 has Frattini factor [ 192, 1543 ].
8269 - 8275 have Frattini factor [ 288, 1024 ].
8276 - 8278 have Frattini factor [ 288, 1025 ].
8279 - 8283 have Frattini factor [ 288, 1026 ].
8284 - 8300 have Frattini factor [ 288, 1027 ].
8301 - 8355 have Frattini factor [ 288, 1028 ].
8356 - 8360 have Frattini factor [ 288, 1029 ].
8361 - 8373 have Frattini factor [ 288, 1030 ].
8374 - 8420 have Frattini factor [ 288, 1031 ].
8421 - 8432 have Frattini factor [ 288, 1032 ].
8433 - 8453 have Frattini factor [ 288, 1033 ].
8454 - 8474 have Frattini factor [ 288, 1034 ].
8475 - 8481 have Frattini factor [ 288, 1035 ].
8482 - 8488 have Frattini factor [ 288, 1036 ].
8489 - 8509 have Frattini factor [ 288, 1037 ].
8510 - 8514 have Frattini factor [ 288, 1038 ].
8515 - 8535 have Frattini factor [ 288, 1039 ].
8536 - 8590 have Frattini factor [ 288, 1040 ].
8591 - 8599 have Frattini factor [ 288, 1041 ].
8600 - 8604 have Frattini factor [ 288, 1042 ].
8605 - 8624 have Frattini factor [ 288, 1043 ].
8625 - 8644 have Frattini factor [ 288, 1044 ].
8645 - 8651 have Frattini factor [ 288, 1045 ].
8652 - 8681 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.