Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9759 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,13)(10,15)(12,17)(16,18)(19,20)(21,25,30,40)(22,32)(23,37)(24,27)(26,35)(28,34)(29,43,33,47)(31,46,44,39)(36,48)(38,49,41,45)(42,50);(3,5,6,13,7,12,15,18)(4,8,10,11,9,16,14,17)(19,21)(20,25,27,40)(22,33)(23,31)(24,30)(26,41,42,38)(28,39,37,46)(29,36)(32,47,48,43)(34,44)(35,49)(45,50);(4,9)(5,12)(10,14)(13,18)(19,22,34,50,24,36,23,35)(20,26,37,48,27,42,28,32)(21,29,44,49,30,33,31,45)(25,38,46,47,40,41,39,43)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112736);