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×#41801 |
| 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 | |
4486 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);(3,4,7,10)(5,8,6,9)(12,16)(17,19)(18,23)(24,25)(27,28,32,33)(29,36,34,30)(31,35,39,40)(37,42,41,38);(12,16)(14,22)(17,19)(20,26)(28,33)(29,34)(35,40)(37,41);(3,4)(5,8)(6,9)(7,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,29,32,34)(28,30,33,36)(31,37,39,41)(35,38,40,42);(3,5,7,6)(4,8,10,9)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30)(28,34)(29,33)(31,38)(32,36)(35,41)(37,40)(39,42);(3,6,7,5)(4,9,10,8)(11,14,15,22)(12,18,16,23)(13,20,21,26)(17,24,19,25)(27,31,32,39)(28,35,33,40)(29,37,34,41)(30,38,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546995);