Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#8158 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1036 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)(5,8)(6,10)(7,9)(11,15)(12,14)(13,16)(17,18)(19,20,23,27,24,28,35,41)(21,29,32,43,31,25,36,38)(22,26,33,39,34,40,46,49)(30,42,45,50,44,37,47,48);(3,5,6,11,7,12,13,17)(4,8,10,15,9,14,16,18)(19,21)(20,25,28,29)(22,30)(23,32)(24,31)(26,37,40,42)(27,38,41,43)(33,45)(34,44)(35,36)(39,48,49,50)(46,47);(4,9)(8,14)(10,16)(15,18)(19,22,23,33,24,34,35,46)(20,26,27,39,28,40,41,49)(21,30,32,45,31,44,36,47)(25,37,38,48,29,42,43,50)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89234);