Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2663 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);(3,4,10,16)(5,17,22,36)(6,23,28,40)(7,29,32,13)(8,15,9,14)(11,21,37,20)(12,26,41,27)(18,34,48,51)(19,42,49,45)(24,53,47,38)(25,52,57,63)(30,35,50,61)(31,43,33,44)(39,56,58,55)(46,60,65,66)(54,64,59,62);(3,5,10,22)(4,11,16,37)(6,24,28,47)(7,30,32,50)(8,21,9,20)(12,38,41,53)(13,42,29,45)(14,36,15,17)(18,27,48,26)(19,33,49,31)(23,51,40,34)(25,46,57,65)(35,44,61,43)(39,60,58,66)(52,62,63,64)(54,56,59,55);(3,6)(4,12)(5,18)(7,25)(8,26)(9,27)(10,28)(11,34)(13,39)(14,40)(15,23)(16,41)(17,38)(19,46)(20,24)(21,47)(22,48)(29,58)(30,59)(31,55)(32,57)(33,56)(35,60)(36,53)(37,51)(42,64)(43,52)(44,63)(45,62)(49,65)(50,54)(61,66);(3,7)(4,13)(5,19)(6,25)(8,33)(9,31)(10,32)(11,35)(12,39)(14,44)(15,43)(16,29)(17,45)(18,46)(20,50)(21,30)(22,49)(23,52)(24,54)(26,56)(27,55)(28,57)(34,60)(36,42)(37,61)(38,62)(40,63)(41,58)(47,59)(48,65)(51,66)(53,64)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407957);