Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C2×C5×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×#41 ◁ G | |
4746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,21)(12,22,25,35)(13,18,26,32)(14,20,15,19)(17,29,31,24)(23,36,37,41)(27,34,28,33)(30,39,40,38);(10,12)(11,17)(13,23)(14,24)(15,29)(16,25)(18,30)(19,22)(20,35)(21,31)(26,37)(27,38)(28,39)(32,40)(33,36)(34,41);(7,9)(10,13)(11,18)(12,23)(14,27)(15,28)(16,26)(17,30)(19,33)(20,34)(21,32)(22,36)(24,38)(25,37)(29,39)(31,40)(35,41);(13,26)(18,32)(23,37)(27,28)(30,40)(33,34)(36,41)(38,39);(10,14,16,15)(11,19,21,20)(12,24,25,29)(13,27,26,28)(17,22,31,35)(18,33,32,34)(23,38,37,39)(30,36,40,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53358);