Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
552 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)(6,7)(8,9)(10,11)(12,20)(13,24)(14,29)(15,18)(16,21)(17,28)(19,27)(22,40)(23,37)(25,35)(26,36)(30,33)(31,39)(32,41)(34,38);(10,12,15,23)(11,16,18,31)(13,25,26,41)(14,30,27,22)(17,33,34,40)(19,35,29,32)(20,36,37,24)(21,38,39,28);(7,9)(13,26)(16,31)(17,34)(21,39)(24,36)(25,41)(28,38)(32,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6305);