Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20884 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,20,16,19,14,28)(12,21,23,17,24,33,29,32)(13,25,26,18,27,35,30,34)(22,31,38,41,39,40,37,36);(7,9)(10,12,15,23,16,24,14,29)(11,17,20,33,19,32,28,21)(13,22,26,38,27,39,30,37)(18,31,35,41,34,40,25,36);(10,13)(11,18)(12,22)(14,30)(15,26)(16,27)(17,31)(19,34)(20,35)(21,36)(23,38)(24,39)(25,28)(29,37)(32,40)(33,41);(11,19)(17,32)(18,34)(20,28)(21,33)(25,35)(31,40)(36,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13872);