Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#8796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #382 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #125 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
520 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);(3,4,7,13,9,14,8,12)(5,11,6,10,16,17,15,18)(19,20,24,30,25,28,23,29)(21,31,33,43,34,26,38,41)(22,35,36,42,37,44,39,27)(32,46,48,45,49,50,47,40);(3,5,8,15,9,16,7,6)(4,10,13,17,14,18,12,11)(19,21,23,33,25,34,24,38)(20,26,29,41,28,31,30,43)(22,32,36,47,37,49,39,48)(27,40,35,50,42,45,44,46);(3,6,7,16,9,15,8,5)(4,11,12,18,14,17,13,10)(19,22)(20,27,28,42)(21,32)(23,36)(24,39)(25,37)(26,40,31,45)(29,35,30,44)(33,47)(34,49)(38,48)(41,50,43,46)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98448);