Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8783 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #382 |
| prime radical | C24 | #117 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
606 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);(3,4,7,10,6,9,14,17)(5,11,13,18,12,8,15,16)(19,20)(21,29,32,25)(22,33,35,48)(23,28)(24,27)(26,34,39,49)(30,43)(31,42,46,38)(36,40)(37,44)(41,50)(45,47);(3,5)(4,8)(6,12)(7,13)(9,11)(10,16)(14,15)(17,18)(19,21,23,31,24,32,36,46)(20,25,28,38,27,29,40,42)(22,30,34,44,35,45,49,50)(26,37,33,47,39,41,48,43);(5,12)(8,11)(13,15)(16,18)(19,22)(20,26,27,39)(21,30)(23,34)(24,35)(25,37,29,41)(28,33,40,48)(31,44)(32,45)(36,49)(38,47,42,43)(46,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98336);