Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,14,19,15,18,27,32)(12,16,22,30,21,29,37,40)(13,23,24,33,25,17,26,31)(20,34,36,41,35,28,38,39);(6,7,8,9)(10,12)(11,16)(13,20)(14,22)(15,21)(17,28)(18,29)(19,30)(23,34)(24,36)(25,35)(26,38)(27,37)(31,39)(32,40)(33,41);(10,13,15,25)(11,17,18,23)(12,20,21,35)(14,26,27,24)(16,28,29,34)(19,33,32,31)(22,38,37,36)(30,41,40,39);(12,21)(16,29)(20,35)(22,37)(28,34)(30,40)(36,38)(39,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13627);