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×#33851 |
| 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 | |
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)(11,16)(12,20)(13,25)(14,28)(17,21)(18,37)(19,33)(23,24)(26,30)(27,39)(29,31)(32,38)(35,41)(36,40);(10,11,15,19)(12,21,24,40)(13,26,29,37)(14,30,28,18)(16,31,33,25)(17,34,36,22)(20,32,23,41)(27,38,39,35);(7,9);(10,12,15,24)(11,17,19,36)(13,27,29,39)(14,20,28,23)(16,32,33,41)(18,38,30,35)(21,37,40,26)(22,31,34,25)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26839);