Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(11,18)(12,24)(13,29)(14,32)(15,16)(19,36)(20,23)(21,30)(22,33)(26,34)(27,31)(28,40)(35,37)(39,41);(6,7)(8,9)(10,11,17,23)(12,25,28,38)(13,30,14,33)(15,22,16,21)(18,32,20,29)(19,27,36,26)(24,37,40,41)(31,39,34,35);(7,9)(10,12)(11,19)(13,31)(14,34)(15,27)(16,26)(17,28)(18,35)(20,39)(21,38)(22,25)(23,36)(24,32)(29,40)(30,41)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6325);