Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2250 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)(7,13)(8,30)(9,14)(10,15)(11,19)(12,25)(17,45)(18,43)(20,34)(21,44)(23,32)(24,53)(26,37)(27,41)(28,39)(29,40)(31,38)(33,50)(35,47)(36,55)(42,48)(46,65)(49,62)(51,60)(52,61)(54,58)(56,63)(57,64)(59,66);(3,5,8,20,10,19,9,21)(4,11,14,34,15,16,30,44)(6,23,25,17,27,48,26,47)(7,18,28,51,29,50,31,52)(12,35,22,42,37,45,41,32)(13,33,38,60,40,43,39,61)(24,54,55,46,56,66,57,49)(36,62,53,59,64,65,63,58);(3,6,10,27)(4,12,15,37)(5,17,19,47)(7,24,29,56)(8,26,9,25)(11,32,16,42)(13,36,40,64)(14,41,30,22)(18,49,50,46)(20,23,21,48)(28,57,31,55)(33,59,43,58)(34,45,44,35)(38,63,39,53)(51,66,52,54)(60,62,61,65);(3,7,10,29)(4,13,15,40)(5,18,19,50)(6,24,27,56)(8,28,9,31)(11,33,16,43)(12,36,37,64)(14,38,30,39)(17,46,47,49)(20,51,21,52)(22,53,41,63)(23,54,48,66)(25,55,26,57)(32,58,42,59)(34,60,44,61)(35,62,45,65)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340367);