Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C4×#2950 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3048 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);(5,6,8,7)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,31)(24,37)(26,29)(28,35)(30,36)(34,40)(38,39);(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,31,39)(18,32,25,19)(24,38,36,35)(30,40,37,33);(5,7,8,6)(9,12,14,23)(10,16,17,31)(11,20,22,34)(13,24,27,36)(15,28,29,39)(18,30,25,37)(19,33,32,40)(21,35,26,38);(10,17)(12,23)(15,29)(18,25)(19,32)(20,34)(24,36)(35,38)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277275);