Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1068 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);(3,4,7,11,8,10,17,5)(6,13,15,18,14,9,16,12)(19,20,24,28,25,30,23,29)(21,31,34,41,35,44,33,26)(22,36,38,27,39,42,37,43)(32,40,47,49,48,50,46,45);(3,5,7,10,8,11,17,4)(6,12,16,13,14,18,15,9)(19,21)(20,26,30,41)(22,32)(23,33)(24,34)(25,35)(27,40,43,50)(28,31,29,44)(36,45,42,49)(37,46)(38,47)(39,48);(3,6,8,14)(4,9,10,13)(5,12,11,18)(7,16,17,15)(19,22,25,39)(20,27,30,43)(21,32,35,48)(23,38,24,37)(26,40,41,50)(28,36,29,42)(31,45,44,49)(33,47,34,46)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106762);