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×#47525 |
| 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 | |
4546 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);(3,4,7,6)(5,9,10,8)(12,16)(17,19)(18,23)(24,25)(27,28)(29,31)(30,34)(32,40)(33,36)(35,41)(37,42)(38,39);(12,16)(14,22)(17,19)(20,26)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(3,5,7,10)(4,8,6,9)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,6,7,4)(5,9,10,8)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,31,33,34)(28,29,36,30)(32,39,41,42)(35,37,40,38);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,32,33,41)(28,35,36,40)(29,37,30,38)(31,39,34,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608124);