Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#23893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4)(5,16)(6,22)(7,28)(8,32)(9,14)(10,15)(11,19)(12,26)(13,29)(17,45)(18,34)(20,36)(21,35)(23,43)(24,54)(25,40)(27,39)(30,42)(31,41)(33,47)(37,51)(38,57)(44,48)(46,63)(49,60)(50,59)(52,55)(53,58)(56,62)(61,66)(64,65);(3,5)(4,11)(6,23)(7,18)(8,20)(9,19)(10,21)(12,37)(13,34)(14,35)(15,36)(16,32)(17,25)(22,33)(24,53)(26,47)(27,51)(28,44)(29,50)(30,48)(31,49)(38,61)(39,45)(40,43)(41,59)(42,60)(46,56)(52,58)(54,63)(55,66)(57,65)(62,64);(3,6)(4,12)(5,17)(7,24)(8,25)(9,26)(10,27)(11,33)(13,38)(14,39)(15,40)(16,43)(18,46)(19,23)(20,51)(21,47)(22,32)(28,52)(29,55)(30,56)(31,57)(34,58)(35,37)(36,45)(41,62)(42,54)(44,63)(48,65)(49,66)(50,53)(59,61)(60,64);(3,7,10,31)(4,13,15,42)(5,18,21,49)(6,24,27,57)(8,30,9,29)(11,34,36,60)(12,38,40,54)(14,41,32,28)(16,44,35,59)(17,46,47,66)(19,50,20,48)(22,52,39,62)(23,53,51,65)(25,56,26,55)(33,58,45,64)(37,61,43,63)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344787);