Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#2211 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C22×D8 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4);(6,8)(9,10)(11,19)(12,15)(13,25)(14,28)(16,21)(17,27)(18,24)(20,35)(22,39)(23,36)(26,34)(29,33)(30,37)(31,38)(32,40);(5,6,7,8)(10,15)(16,30)(17,31)(18,28)(19,35)(25,36)(32,34)(33,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,33,31,39)(18,34,28,32)(19,36,35,25)(21,38,37,27)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403007);