Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C22×C5×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
718 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)(6,7,9,8)(10,11)(12,17)(13,16)(14,27)(15,18)(19,24)(20,25)(21,33)(22,28)(23,37)(26,29)(30,38)(31,36)(32,35)(34,40)(39,41);(11,16)(13,24)(14,25)(17,28)(18,29)(19,20)(22,36)(23,35)(26,38)(27,30)(31,32)(33,40)(34,41)(37,39);(6,8,9,7)(10,12)(11,17)(13,22)(14,23)(15,21)(16,28)(18,33)(19,31)(20,32)(24,36)(25,35)(26,34)(27,37)(29,40)(30,39)(38,41);(11,18)(12,21)(16,29)(19,30)(20,27)(22,34)(23,35)(36,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26444);