Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#12034 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#107 |
| prime radical | C24 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,30)(22,27)(23,37)(24,39)(25,29)(26,33)(28,31)(32,43)(34,49)(35,45)(36,42)(38,50)(40,47)(41,48)(44,46);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,31,46,24,34,38,48,25,35,36,47,22,32,23,33)(20,26,37,43,27,40,42,45,29,41,50,49,39,44,28,30);(5,11)(8,14)(12,17)(15,18)(19,22,25,24)(20,27,29,39)(21,32,35,34)(23,36,38,31)(26,40,41,44)(28,37,42,50)(30,43,45,49)(33,47,48,46)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390624);