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×#52754 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4646 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);(3,4,6,8)(5,9,10,7)(14,22)(17,24)(20,26)(23,25)(28,33)(29,36)(31,39)(38,41);(5,10)(7,9)(11,12,15,18)(13,16,21,19)(14,17,22,24)(20,23,26,25)(27,28,32,33)(29,30,36,34)(31,35,39,40)(37,38,42,41);(3,5,6,10)(4,7,8,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26)(27,29,32,36)(28,34,33,30)(31,37,39,42)(35,41,40,38);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25)(27,30,32,34)(28,29,33,36)(31,38,39,41)(35,37,40,42);(11,14)(12,17)(13,20)(15,22)(16,23)(18,24)(19,25)(21,26)(27,31)(28,35)(29,37)(30,38)(32,39)(33,40)(34,41)(36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999722);