Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47311 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12,16,19)(13,14,17,22)(15,18,24,26)(20,21,23,25)(27,28,32,37)(29,34,40,33)(30,35,36,31)(38,39,41,42);(11,13,16,17)(12,14,19,22)(15,23,24,20)(18,25,26,21)(27,29,32,40)(28,33,37,34)(30,41,36,38)(31,42,35,39);(3,5,7,10)(4,8,6,9)(11,13)(12,14)(15,20)(16,17)(18,21)(19,22)(23,24)(25,26)(27,29)(28,34)(30,41)(31,39)(32,40)(33,37)(35,42)(36,38);(3,6,7,4)(5,9,10,8)(11,14)(12,17)(13,19)(15,21)(16,22)(18,23)(20,26)(24,25)(27,30,32,36)(28,35,37,31)(29,38,40,41)(33,42,34,39);(11,15)(12,18)(13,20)(14,21)(16,24)(17,23)(19,26)(22,25)(27,31,32,35)(28,36,37,30)(29,39,40,42)(33,41,34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608616);