Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33846 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ 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)(8,9)(10,11,15,19)(12,20,24,38)(13,25,28,41)(14,29,27,17)(16,31,33,22)(18,35,37,26)(21,34,39,32)(23,40,30,36);(10,12)(11,16)(13,21)(14,30)(15,24)(17,34)(18,36)(19,33)(20,25)(22,26)(23,27)(28,39)(29,32)(31,35)(37,40)(38,41);(7,9);(11,17)(12,21)(13,26)(14,27)(16,20)(18,37)(19,29)(22,23)(24,39)(28,35)(30,31)(32,40)(33,38)(34,36)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26834);