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×#55647 |
| 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 | |
4534 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)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(28,32)(29,35)(30,38)(40,42);(5,9)(8,10)(12,14)(17,19)(18,22)(24,25)(27,28,31,32)(29,36,35,33)(30,34,38,39)(37,42,41,40);(3,4,6,7)(5,8,9,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39)(41,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,29,31,35)(28,33,32,36)(30,37,38,41)(34,40,39,42);(3,5)(4,8)(6,9)(7,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,30,31,38)(28,34,32,39)(29,37,35,41)(33,40,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008080);