Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2653 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)(5,17)(6,23)(7,13)(8,15)(9,14)(10,16)(11,20)(12,25)(18,36)(19,45)(21,35)(22,34)(24,53)(26,43)(27,58)(28,41)(29,40)(30,42)(31,38)(32,48)(33,49)(37,55)(39,50)(44,47)(46,59)(51,60)(52,56)(54,62)(57,63)(61,65)(64,66);(3,5,10,22)(4,11,16,35)(6,18,26,48)(7,27,30,49)(8,21,9,20)(12,36,38,32)(13,39,42,45)(14,34,15,17)(19,29,50,28)(23,44,43,60)(24,54,56,66)(25,51,31,47)(33,41,58,40)(37,62,63,64)(46,57,65,55)(52,59,53,61);(3,6)(4,12)(5,18)(7,24)(8,25)(9,31)(10,26)(11,32)(13,37)(14,23)(15,43)(16,38)(17,44)(19,46)(20,47)(21,51)(22,48)(27,54)(28,55)(29,57)(30,56)(33,59)(34,60)(35,36)(39,64)(40,53)(41,52)(42,63)(45,62)(49,66)(50,65)(58,61);(3,7,10,30)(4,13,16,42)(5,19,22,50)(6,24,26,56)(8,29,9,28)(11,33,35,58)(12,37,38,63)(14,41,15,40)(17,45,34,39)(18,46,48,65)(20,49,21,27)(23,52,43,53)(25,57,31,55)(32,59,36,61)(44,62,60,64)(47,66,51,54)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407949);