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×#33459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,18)(13,21)(14,28)(15,19)(16,22)(17,24)(20,30)(23,37)(25,36)(26,34)(27,39)(29,35)(31,38)(32,41)(33,40);(10,12)(11,17)(13,23)(14,29)(15,26)(16,25)(18,28)(19,32)(20,34)(21,35)(22,33)(24,30)(27,40)(31,41)(36,38)(37,39);(6,8,9,7)(10,12)(11,18)(13,26)(14,24)(15,23)(16,25)(17,28)(19,37)(20,35)(21,34)(22,36)(27,41)(29,30)(31,40)(32,39)(33,38);(10,13,16,15)(11,19,22,21)(12,23,25,26)(14,27,31,30)(17,32,33,35)(18,34,36,37)(20,38,39,28)(24,29,40,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26447);