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×#8795 |
| 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);(4,9)(5,10)(6,7)(11,12)(13,21,25,17)(14,26,29,40)(15,20)(16,19)(18,28,34,37)(22,30)(23,38)(24,35,36,32)(27,41)(31,39)(33,42);(3,4,6,9,8,10,7,5)(11,13,22,36,16,25,15,24)(12,17,30,35,19,21,20,32)(14,27,37,39,29,23,28,42)(18,33,40,41,34,31,26,38);(3,5,6,4,8,9,7,10)(11,14)(12,18,19,34)(13,23)(15,28)(16,29)(17,31,21,33)(20,26,30,40)(22,37)(24,39)(25,27)(32,41,35,38)(36,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98423);