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×#54782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4810 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);(3,4,6,7)(5,9,10,8)(11,12)(13,19)(14,22)(15,21)(16,17)(18,23)(20,25)(24,26)(27,28)(29,34)(30,36)(31,33)(32,35)(37,41)(38,42)(39,40);(4,7)(8,9)(12,17)(14,23)(15,19)(20,24)(27,29,33,30)(28,34,31,36)(32,38,40,37)(35,42,39,41);(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,24)(16,21)(17,19)(18,26)(20,23)(22,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,4)(5,8)(6,7)(9,10)(11,14)(12,18)(13,20)(15,25)(16,23)(17,22)(19,26)(21,24)(27,31,33,28)(29,36,30,34)(32,39,40,35)(37,42,38,41);(3,5)(4,8)(6,10)(7,9)(11,15)(12,13)(14,25)(16,19)(17,21)(18,20)(22,24)(23,26)(27,32,33,40)(28,35,31,39)(29,37,30,38)(34,41,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006980);