Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#8792 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #382 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ 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)(5,10)(6,15)(7,17)(8,12)(9,13)(11,16)(14,18)(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);(3,5,7,13,8,14,16,6)(4,9,11,10,12,15,17,18)(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);(3,6,16,14,8,13,7,5)(4,10,17,9,12,18,11,15)(19,22,24,35)(20,26)(21,30,32,45)(23,34,36,49)(25,37)(27,39)(28,33)(29,41)(31,44,46,50)(38,47)(40,48)(42,43)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98417);