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×#33848 |
| 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 | |
696 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)(12,20,24,39)(13,25,28,38)(14,29,27,17)(16,31,33,22)(18,35,37,26)(21,34,40,32)(23,41,30,36);(10,12,15,24)(11,16,19,33)(13,21,28,40)(14,30,27,23)(17,34,29,32)(18,36,37,41)(20,38,39,25)(22,35,31,26);(7,9);(11,17)(12,21)(13,26)(14,27)(16,20)(18,37)(19,29)(22,23)(24,40)(28,35)(30,31)(32,41)(33,39)(34,36)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26836);