Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #397 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
594 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)(12,21)(13,25)(14,30)(15,19)(16,20)(17,23)(18,28)(22,35)(24,33)(26,40)(27,36)(29,39)(31,34)(32,41)(37,38);(10,12)(11,17)(13,26,28,41)(14,31)(15,23)(16,24)(18,35,25,37)(19,33)(20,34)(21,30)(22,29,38,27)(32,36,40,39);(10,13,14,27,16,28,15,29)(11,18,19,36,20,25,30,39)(12,22,23,26,24,38,31,41)(17,32,21,37,34,40,33,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4269);