Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #381 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1162 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)(10,11,14,19,15,18,28,33)(12,16,22,31,21,30,26,34)(13,23,24,35,25,17,27,32)(20,36,38,41,37,29,39,40);(10,12)(11,16,18,30)(13,20)(14,26)(15,21)(17,29,23,36)(19,34,33,31)(22,28)(24,39)(25,37)(27,38)(32,41,35,40);(7,9)(10,13,15,25)(11,17,18,23)(12,20,21,37)(14,27,28,24)(16,29,30,36)(19,35,33,32)(22,39,26,38)(31,41,34,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3758);