Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54729 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,9)(5,6,10,8)(12,16)(13,19)(18,23)(20,25)(27,28,32,35)(29,30,37,33)(31,34,39,41)(36,38,42,40);(3,5,7,10)(4,8,9,6)(12,16)(17,22)(18,23)(24,26)(31,39)(34,41)(36,42)(38,40);(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29,32,37)(28,33,35,30)(31,36,39,42)(34,40,41,38);(3,5)(4,6)(7,10)(8,9)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,30,32,33)(28,29,35,37)(31,38,39,40)(34,36,41,42);(3,6)(4,5)(7,8)(9,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,31,32,39)(28,34,35,41)(29,36,37,42)(30,38,33,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010973);