Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18685 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3544 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(8,10)(11,12,16,20)(13,21,24,34)(14,18,25,32)(15,27,29,19)(17,28,31,23)(22,35,36,42)(26,39,38,33)(30,40,41,37);(5,9)(7,10)(11,13,16,24)(12,17,20,31)(14,22,25,36)(15,28,29,23)(18,30,32,41)(19,34,27,21)(26,40,38,37)(33,42,39,35);(3,4)(5,10)(6,8)(7,9)(11,14)(12,18)(13,22)(15,26)(16,25)(17,30)(19,33)(20,32)(21,35)(23,37)(24,36)(27,39)(28,40)(29,38)(31,41)(34,42);(14,25)(18,32)(22,36)(26,38)(30,41)(33,39)(35,42)(37,40)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255477);