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×#52521 |
| 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 | |
4652 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);(5,9)(8,10)(11,12,16,17)(13,18,19,14)(15,23,24,20)(21,22,26,25)(27,28)(29,34)(30,31)(32,40)(33,36)(35,41)(37,38)(39,42);(3,4)(5,8)(6,7)(9,10)(12,17)(14,18)(20,23)(22,25)(27,29,33,31)(28,34,36,30)(32,37,41,39)(35,42,40,38);(4,7)(8,10)(11,13)(12,18)(14,17)(15,21)(16,19)(20,25)(22,23)(24,26)(27,30)(28,29)(31,36)(32,38)(33,34)(35,37)(39,40)(41,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,18)(12,19,17,13)(15,22,24,25)(20,26,23,21)(27,31,33,29)(28,34,36,30)(32,39,41,37)(35,42,40,38);(3,5)(4,8)(6,9)(7,10)(11,15,16,24)(12,20,17,23)(13,21,19,26)(14,22,18,25)(27,32)(28,35)(29,37)(30,38)(31,39)(33,41)(34,42)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999568);