Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#61108 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10152 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)(5,17)(6,22)(7,13)(8,16)(9,15)(10,14)(11,20)(12,24)(18,32)(19,44)(21,40)(23,52)(25,42)(26,57)(27,41)(28,38)(29,34)(30,39)(31,36)(33,49)(35,55)(37,50)(43,47)(45,58)(46,51)(48,59)(53,63)(54,61)(56,62)(60,65)(64,66);(3,5)(4,11)(6,18)(7,26)(8,29)(9,20)(10,21)(12,32)(13,37)(14,34)(15,40)(16,17)(19,27)(22,51)(23,53)(24,47)(25,48)(28,49)(30,50)(31,46)(33,39)(35,60)(36,59)(38,44)(41,57)(42,43)(45,54)(52,66)(55,64)(56,65)(58,62)(61,63);(3,6,10,25)(4,12,14,36)(5,18,21,48)(7,23,28,56)(8,24,9,31)(11,32,34,59)(13,35,38,61)(15,22,16,42)(17,43,40,51)(19,45,50,64)(20,46,29,47)(26,53,49,65)(27,54,30,55)(33,58,57,66)(37,60,44,63)(39,62,41,52);(3,7)(4,13)(5,19)(6,23)(8,30)(9,27)(10,28)(11,33)(12,35)(14,38)(15,41)(16,39)(17,44)(18,45)(20,49)(21,50)(22,52)(24,55)(25,56)(26,29)(31,54)(32,58)(34,57)(36,61)(37,40)(42,62)(43,63)(46,65)(47,53)(48,64)(51,60)(59,66);(4,14)(11,34)(12,36)(13,38)(15,16)(17,40)(22,42)(32,59)(33,57)(35,61)(37,44)(39,41)(43,51)(52,62)(58,66)(60,63)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483367);