Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#2138 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C24 | C2×D8 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2);(3,4);(5,6,7,8)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,22)(18,27)(21,36)(23,37)(24,35)(26,31)(28,39)(30,34)(32,40)(33,38);(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,34)(19,33)(20,25)(22,27)(24,38)(28,40)(29,32)(35,36)(37,39);(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,33,32)(18,25,35,39)(20,36,37,34)(22,26,38,40)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529685);