Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4766 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)(5,9)(14,22)(18,23)(20,26)(24,25)(27,28)(29,33)(30,32)(31,34)(35,36)(37,40)(38,39)(41,42);(4,7)(8,10)(12,16)(17,19)(18,23)(24,25)(28,30)(29,35)(34,38)(37,41);(3,4)(5,8)(6,7)(9,10)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,29,32,35)(28,33,30,36)(31,37,39,41)(34,40,38,42);(3,5)(4,8)(6,9)(7,10)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30,32,28)(29,36,35,33)(31,38,39,34)(37,42,41,40);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,39)(28,34,30,38)(29,37,35,41)(33,40,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999940);