Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4562 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);(4,6)(7,8)(9,10)(11,12,16,20)(13,17,22,31)(14,24,26,34)(15,27,29,19)(18,28,33,25)(21,35,36,42)(23,39,38,32)(30,40,41,37);(3,4,5,6)(11,13)(12,17)(14,21)(15,23)(16,22)(18,30)(19,32)(20,31)(24,35)(25,37)(26,36)(27,39)(28,40)(29,38)(33,41)(34,42);(11,14,16,26)(12,18,20,33)(13,21,22,36)(15,28,29,25)(17,30,31,41)(19,34,27,24)(23,40,38,37)(32,42,39,35);(8,10)(13,22)(17,31)(21,36)(23,38)(30,41)(32,39)(35,42)(37,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536115);