Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(5,11)(6,13)(8,16)(9,14)(12,15)(17,18)(19,20,25,30)(21,31,35,41)(22,36,38,44)(23,29,24,28)(26,33,42,34)(27,39,43,37)(32,40,48,50)(45,47,49,46);(3,4,7,10)(5,8,12,17)(6,14,13,9)(11,18,15,16)(19,21)(20,26)(22,32)(23,33)(24,34)(25,35)(27,40)(28,41)(29,31)(30,42)(36,49)(37,46)(38,48)(39,47)(43,50)(44,45);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,13)(14,16)(19,22)(20,27)(21,32)(23,37)(24,39)(25,38)(26,40)(28,36)(29,44)(30,43)(31,45)(33,46)(34,47)(35,48)(41,49)(42,50);(19,23,25,24)(20,28,30,29)(21,33,35,34)(22,37,38,39)(26,41,42,31)(27,36,43,44)(32,46,48,47)(40,49,50,45)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258583);