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×#55078 |
| 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 | |
4636 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);(11,12)(13,19)(14,16)(15,22)(17,20)(18,23)(21,24)(25,26)(27,28,32,36)(29,30,34,33)(31,39,41,35)(37,42,40,38);(3,4,7,8)(5,6,9,10)(12,14)(13,20)(18,22)(21,26)(27,29,32,34)(28,33,36,30)(31,40,41,37)(35,38,39,42);(4,8)(5,9)(11,13)(12,17)(14,19)(15,21)(16,20)(18,24)(22,25)(23,26)(27,29)(28,30)(31,37)(32,34)(33,36)(35,38)(39,42)(40,41);(3,5)(4,6)(7,9)(8,10)(11,14,16,12)(13,19,20,17)(15,22,23,18)(21,25,26,24)(27,30)(28,34)(29,36)(31,38)(32,33)(35,40)(37,39)(41,42);(3,6)(4,5)(7,10)(8,9)(11,15,16,23)(12,18,14,22)(13,21,20,26)(17,24,19,25)(27,31,32,41)(28,35,36,39)(29,37,34,40)(30,38,33,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009565);