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×C4 |
| direct product | C2×#8791 |
| 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)(5,13)(6,9)(7,17)(8,12)(10,15)(11,14)(16,18)(19,20)(21,29,33,25)(22,34,36,49)(23,28)(24,27)(26,35,40,44)(30,37)(31,45)(32,42,43,39)(38,46)(41,50)(47,48);(3,5,14,16,8,15,7,6)(4,9,17,10,12,18,11,13)(19,21,30,43,24,33,23,32)(20,25,37,42,27,29,28,39)(22,31,44,50,36,47,35,46)(26,38,49,45,40,41,34,48);(3,6,7,15,8,16,14,5)(4,10,11,9,12,13,17,18)(19,22)(20,26,27,40)(21,31)(23,35)(24,36)(25,38,29,41)(28,34,37,49)(30,44)(32,46)(33,47)(39,48,42,45)(43,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98416);