Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1208 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)(5,11)(6,14)(7,10)(8,13)(9,16)(12,18)(15,17)(19,20,25,30)(21,31,34,42)(22,27,38,45)(23,29,24,28)(26,35,41,33)(32,47,48,43)(36,50,49,40)(37,46,39,44);(8,17)(11,18)(19,21,25,34)(20,26,30,41)(22,36,38,49)(23,35,24,33)(27,43,45,47)(28,42,29,31)(32,39,48,37)(40,46,50,44);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,14)(13,16)(19,22,25,38)(20,27,30,45)(21,32,34,48)(23,37,24,39)(26,40,41,50)(28,44,29,46)(31,43,42,47)(33,36,35,49)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142734);