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×#54695 |
| 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 | |
4664 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);(3,4,8,9)(5,7,6,10)(11,12)(13,19)(14,16)(15,22)(17,21)(18,23)(20,24)(25,26)(27,28,33,35)(29,36,30,31)(32,40,41,34)(37,39,38,42);(4,9)(5,6)(12,14)(15,23)(17,19)(20,26)(27,29,33,30)(28,31,35,36)(32,37,41,38)(34,39,40,42);(3,5,8,6)(4,10,9,7)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,30,33,29)(28,31,35,36)(32,38,41,37)(34,39,40,42);(3,6,8,5)(4,7,9,10)(11,14,16,12)(13,17,21,19)(15,22,23,18)(20,24,26,25)(27,31)(28,29)(30,35)(32,39)(33,36)(34,37)(38,40)(41,42);(3,7)(4,5)(6,9)(8,10)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,32,33,41)(28,34,35,40)(29,37,30,38)(31,39,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010790);