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×#57696 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4484 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);(3,4,6,7)(5,9,10,8)(11,12,16,19)(13,14,21,17)(15,18,23,25)(20,22,26,24)(27,28,32,37)(29,34,40,33)(30,35,36,31)(38,39,41,42);(4,7)(5,10)(15,23)(18,25)(20,26)(22,24)(27,29,32,40)(28,33,37,34)(30,41,36,38)(31,42,35,39);(4,7)(8,9)(11,13,16,21)(12,17,19,14)(15,20,23,26)(18,24,25,22)(27,29)(28,34)(30,41)(31,39)(32,40)(33,37)(35,42)(36,38);(3,4)(5,8)(6,7)(9,10)(11,14,16,17)(12,13,19,21)(15,22,23,24)(18,20,25,26)(27,30,32,36)(28,35,37,31)(29,38,40,41)(33,42,34,39);(3,5,6,10)(4,8,7,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,35)(28,36,37,30)(29,39,40,42)(33,41,34,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008988);