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×#47529 |
| 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 | |
4382 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);(4,7)(8,10)(11,12,16,19)(13,15,18,24)(14,17,23,26)(20,22,21,25)(27,28,33,31)(29,36,34,30)(32,39,41,35)(37,38,40,42);(13,18)(15,24)(20,21)(22,25)(27,29,33,34)(28,30,31,36)(32,40,41,37)(35,42,39,38);(3,4)(5,8)(6,7)(9,10)(11,13,16,18)(12,15,19,24)(14,21,23,20)(17,25,26,22)(27,30,33,36)(28,34,31,29)(32,38,41,42)(35,40,39,37);(11,14)(12,17)(13,20)(15,22)(16,23)(18,21)(19,26)(24,25)(27,31,33,28)(29,30,34,36)(32,39,41,35)(37,38,40,42);(3,5,6,9)(4,8,7,10)(11,15)(12,18)(13,19)(14,22)(16,24)(17,21)(20,26)(23,25)(27,32,33,41)(28,35,31,39)(29,37,34,40)(30,38,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608100);