Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51484 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,5)(6,9,10,8)(11,12,16,17)(13,19,20,14)(15,23,24,18)(21,22,26,25)(27,28,32,35)(29,36,37,33)(30,31,34,41)(38,39,40,42);(3,5,7,4)(6,9,10,8)(12,17)(13,20)(15,24)(22,25)(29,37)(30,34)(31,41)(33,36);(3,6)(4,8)(5,9)(7,10)(11,13,16,20)(12,14,17,19)(15,21,24,26)(18,22,23,25)(27,29,32,37)(28,33,35,36)(30,40,34,38)(31,39,41,42);(11,14)(12,13)(15,22)(16,19)(17,20)(18,21)(23,26)(24,25)(27,30,32,34)(28,31,35,41)(29,38,37,40)(33,42,36,39);(11,15,16,24)(12,18,17,23)(13,21,20,26)(14,22,19,25)(27,31)(28,34)(29,39)(30,35)(32,41)(33,38)(36,40)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614550);