Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
4390 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)(5,17,22,33)(6,23,26,39)(7,13,28,38)(8,15,9,14)(11,21,34,20)(12,29,36,25)(18,43,46,31)(19,42,48,59)(24,52,55,51)(27,40,30,37)(32,50,57,47)(35,53,60,54)(41,49,58,45)(44,56,65,62)(61,63,66,64);(3,5,9,21,10,22,8,20)(4,11,15,33,16,34,14,17)(6,18,29,49,26,46,25,45)(7,19,30,50,28,48,27,47)(12,31,39,58,36,43,23,41)(13,32,40,59,38,57,37,42)(24,44,54,64,55,65,53,63)(35,56,52,66,60,62,51,61);(3,6)(4,12)(5,18)(7,24)(8,25)(9,29)(10,26)(11,31)(13,35)(14,23)(15,39)(16,36)(17,41)(19,44)(20,45)(21,49)(22,46)(27,53)(28,55)(30,54)(32,56)(33,58)(34,43)(37,51)(38,60)(40,52)(42,61)(47,63)(48,65)(50,64)(57,62)(59,66);(3,7)(4,13)(5,19)(6,24)(8,27)(9,30)(10,28)(11,32)(12,35)(14,37)(15,40)(16,38)(17,42)(18,44)(20,47)(21,50)(22,48)(23,51)(25,53)(26,55)(29,54)(31,56)(33,59)(34,57)(36,60)(39,52)(41,61)(43,62)(45,63)(46,65)(49,64)(58,66)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536245);