Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37125 |
| 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 | |
4648 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);(5,9)(7,10)(14,21)(18,23)(20,25)(24,26)(27,28)(29,35)(30,34)(31,32)(33,37)(36,41)(38,39)(40,42);(3,4,6,8)(5,7,9,10)(12,16)(13,19)(18,23)(20,25)(28,32)(33,35)(34,39)(40,41);(3,5)(4,7)(6,9)(8,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,28)(29,33)(30,34)(31,32)(35,37)(36,40)(38,39)(41,42);(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,29,31,37)(28,33,32,35)(30,36,38,42)(34,40,39,41);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,30,31,38)(28,34,32,39)(29,36,37,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541680);