Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#60472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 66 points:
(1,2);(3,4,10,16)(5,17,22,33)(6,23,26,40)(7,13,27,37)(8,15,9,14)(11,21,34,20)(12,30,36,25)(18,43,46,31)(19,42,47,57)(24,51,52,60)(28,39,29,38)(32,49,56,48)(35,54,59,53)(41,50,58,45)(44,62,63,55)(61,65,66,64);(3,5,9,21,10,22,8,20)(4,11,15,33,16,34,14,17)(6,18,30,50,26,46,25,45)(7,19,29,49,27,47,28,48)(12,31,40,58,36,43,23,41)(13,32,39,57,37,56,38,42)(24,44,54,65,52,63,53,64)(35,55,60,66,59,62,51,61);(3,6)(4,12)(5,18)(7,24)(8,25)(9,30)(10,26)(11,31)(13,35)(14,23)(15,40)(16,36)(17,41)(19,44)(20,45)(21,50)(22,46)(27,52)(28,53)(29,54)(32,55)(33,58)(34,43)(37,59)(38,51)(39,60)(42,61)(47,63)(48,64)(49,65)(56,62)(57,66);(3,7)(4,13)(5,19)(6,24)(8,28)(9,29)(10,27)(11,32)(12,35)(14,38)(15,39)(16,37)(17,42)(18,44)(20,48)(21,49)(22,47)(23,51)(25,53)(26,52)(30,54)(31,55)(33,57)(34,56)(36,59)(40,60)(41,61)(43,62)(45,64)(46,63)(50,65)(58,66);(7,27)(13,37)(19,47)(24,52)(28,29)(32,56)(35,59)(38,39)(42,57)(44,63)(48,49)(51,60)(53,54)(55,62)(61,66)(64,65)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482110);