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×#47530 |
| 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 | |
4578 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);(4,7)(8,10)(11,12,16,19)(13,15,18,24)(14,17,23,26)(20,22,21,25)(28,33)(29,36)(31,39)(38,41);(13,18)(15,24)(20,21)(22,25)(27,28,32,33)(29,30,36,34)(31,40,39,35)(37,41,42,38);(3,4)(5,8)(6,7)(9,10)(11,13,16,18)(12,15,19,24)(14,21,23,20)(17,25,26,22)(27,29,32,36)(28,34,33,30)(31,37,39,42)(35,41,40,38);(11,14)(12,17)(13,20)(15,22)(16,23)(18,21)(19,26)(24,25)(27,30,32,34)(28,29,33,36)(31,38,39,41)(35,37,40,42);(3,5,6,9)(4,8,7,10)(11,15)(12,18)(13,19)(14,22)(16,24)(17,21)(20,26)(23,25)(27,31,32,39)(28,35,33,40)(29,37,36,42)(30,38,34,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608116);