Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33844 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3287 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ 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)(8,9)(10,11,16,20)(12,21,22,40)(13,26,18,38)(14,29,32,28)(15,33,31,19)(17,35,36,24)(23,41,37,30)(25,27,34,39);(10,12,16,22)(11,17,20,36)(13,27,18,39)(14,30,32,41)(15,34,31,25)(19,23,33,37)(21,28,40,29)(24,38,35,26);(7,9)(12,22)(17,36)(21,40)(23,37)(24,35)(25,34)(27,39)(30,41);(10,13,16,18)(11,14,20,32)(12,23,22,37)(15,29,31,28)(17,25,36,34)(19,38,33,26)(21,35,40,24)(27,41,39,30)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26832);