Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1824 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);(3,4)(5,12)(6,10)(7,16)(8,11)(9,15)(13,18)(14,17)(19,20,25,29)(21,30,34,45)(22,27,36,43)(23,28,24,38)(26,32,40,33)(31,44,47,49)(35,41,37,42)(39,48,50,46);(5,13)(9,17)(12,18)(14,15)(19,21,23,32,25,34,24,33)(20,26,38,45,29,40,28,30)(22,31,37,48,36,47,35,46)(27,39,41,49,43,50,42,44);(3,5,7,15,8,13,6,14)(4,9,10,12,11,17,16,18)(19,22)(20,27)(21,31)(23,37)(24,35)(25,36)(26,39)(28,42)(29,43)(30,44)(32,48)(33,46)(34,47)(38,41)(40,50)(45,49);(3,6,8,7)(4,10,11,16)(5,14,13,15)(9,12,17,18)(19,23,25,24)(20,28,29,38)(21,32,34,33)(22,35,36,37)(26,30,40,45)(27,41,43,42)(31,46,47,48)(39,49,50,44)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264490);