Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11111 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #392 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,22)(13,26)(14,20)(15,19)(16,21)(17,24)(18,27)(23,31)(25,35)(28,36)(29,32)(30,34)(33,40)(37,39)(38,41);(10,12,14,24,16,25,15,29)(11,17,19,22,21,32,20,35)(13,23,27,38,28,40,30,39)(18,33,26,41,34,31,36,37);(10,13)(11,18)(12,23)(14,27)(15,30)(16,28)(17,31)(19,26)(20,36)(21,34)(22,37)(24,38)(25,40)(29,39)(32,33)(35,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4099);