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×#57092 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| 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);(3,4)(5,9)(6,10)(7,8)(12,16)(13,19)(18,23)(21,25)(27,28,32,31)(29,35,36,33)(30,38,39,34)(37,40,42,41);(3,5,7,10)(4,6,8,9)(14,22)(18,23)(21,25)(24,26)(29,36)(33,35)(37,42)(40,41);(4,8)(6,9)(11,12,15,16)(13,20,19,17)(14,18,22,23)(21,26,25,24)(27,29)(28,33)(30,37)(31,35)(32,36)(34,40)(38,41)(39,42);(3,6)(4,5)(7,9)(8,10)(11,13,15,19)(12,17,16,20)(14,21,22,25)(18,24,23,26)(27,30,32,39)(28,34,31,38)(29,37,36,42)(33,40,35,41);(3,5)(4,6)(7,10)(8,9)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31,32,28)(29,35,36,33)(30,38,39,34)(37,41,42,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006782);