Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11288 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
602 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)(10,11,14,19,15,18,27,33)(12,16,22,30,21,29,25,34)(13,17,24,32,23,31,26,35)(20,28,37,39,36,40,38,41);(6,7)(8,9)(10,12)(11,16,18,29)(13,20,23,36)(14,25)(15,21)(17,28)(19,34,33,30)(22,27)(24,38,26,37)(31,40)(32,41)(35,39);(7,9)(10,13)(11,17,18,31)(12,20,21,36)(14,26)(15,23)(16,28)(19,35,33,32)(22,38,25,37)(24,27)(29,40)(30,41)(34,39)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4276);