Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41698 |
| 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);(4,7)(8,10)(11,12,16,20)(13,15,19,17)(14,18,24,26)(21,23,22,25)(27,28)(29,36)(30,34)(31,33)(32,35)(37,42)(38,41)(39,40);(13,19)(15,17)(21,22)(23,25)(27,29,33,30)(28,34,31,36)(32,38,40,37)(35,42,39,41);(3,4,6,7)(5,8,9,10)(11,13,16,19)(12,17,20,15)(14,22,24,21)(18,23,26,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(11,14)(12,18)(13,21)(15,23)(16,24)(17,25)(19,22)(20,26)(27,31,33,28)(29,36,30,34)(32,39,40,35)(37,42,38,41);(3,5,6,9)(4,8,7,10)(11,15,16,17)(12,19,20,13)(14,23,24,25)(18,22,26,21)(27,32)(28,35)(29,37)(30,38)(31,39)(33,40)(34,41)(36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547048);