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×#33836 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
664 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)(13,24)(14,27)(17,34)(18,38)(19,33)(21,37)(22,35)(23,30)(25,29)(28,31)(32,36)(39,40);(10,11,15,19)(12,21,23,34)(13,25,28,38)(14,29,27,18)(16,31,33,24)(17,35,37,22)(20,40,30,32)(26,39,41,36);(7,9);(10,12,15,23)(11,17,19,37)(13,26,28,41)(14,30,27,20)(16,32,33,40)(18,39,29,36)(21,38,34,25)(22,31,35,24)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26824);