Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,10,16)(5,17,22,50)(6,23,27,47)(7,28,33,45)(8,15,9,14)(11,35,39,19)(12,34,42,26)(13,31,46,32)(18,51,54,66)(20,49,21,43)(24,58,53,48)(25,44,30,41)(29,38,56,37)(36,62,65,59)(40,55,64,52)(57,63,61,60);(3,5)(4,11)(6,24)(7,29)(8,20)(9,21)(10,22)(12,40)(13,43)(14,37)(15,38)(16,39)(17,28)(18,26)(19,32)(23,36)(25,55)(27,53)(30,52)(31,35)(33,56)(34,54)(41,48)(42,64)(44,58)(45,50)(46,49)(47,65)(51,57)(59,60)(61,66)(62,63);(3,6)(4,12)(5,18)(7,30)(8,34)(9,26)(10,27)(11,36)(13,44)(14,47)(15,23)(16,42)(17,48)(19,55)(20,24)(21,53)(22,54)(25,33)(28,61)(29,62)(31,60)(32,63)(35,52)(37,40)(38,64)(39,65)(41,46)(43,66)(45,57)(49,51)(50,58)(56,59)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406032);