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×#33853 |
| 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) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(11,16)(12,21)(13,22)(14,23)(17,28)(18,24)(19,34)(20,32)(26,27)(29,39)(30,31)(33,38)(35,41)(37,40);(10,11,15,19)(12,17,21,20)(13,24,26,40)(14,28,30,32)(16,31,34,23)(18,36,37,25)(22,33,27,41)(29,38,39,35);(7,9)(10,12,15,21)(11,17,19,20)(13,22,26,27)(14,23,30,31)(16,32,34,28)(18,35,37,38)(24,33,40,41)(25,39,36,29);(10,13,15,26)(11,18,19,37)(12,22,21,27)(14,29,30,39)(16,33,34,41)(17,35,20,38)(23,25,31,36)(24,32,40,28)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26841);