Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8794 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×#8794 |
| 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 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,8,5)(6,7,10,9)(11,12)(13,21,24,17)(14,25,28,40)(15,20)(16,19)(18,27,33,37)(22,36)(23,35,39,31)(26,41)(29,34)(30,38)(32,42);(3,5,7,10,8,4,9,6)(11,13,15,23,16,24,29,39)(12,17,20,31,19,21,34,35)(14,26,27,42,28,22,37,38)(18,32,25,36,33,30,40,41);(3,6,7,5,8,10,9,4)(11,14)(12,18,19,33)(13,22)(15,27)(16,28)(17,30,21,32)(20,25,34,40)(23,38)(24,26)(29,37)(31,41,35,36)(39,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98422);