Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1360 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)(10,11,16,21,17,22,15,31)(12,23,14,18,25,20,30,34)(13,26,28,37,29,36,32,19)(24,39,40,35,41,33,27,38);(6,7)(8,9)(10,12)(11,18)(13,24)(14,16)(15,30)(17,25)(19,33)(20,21)(22,34)(23,31)(26,38)(27,32)(28,40)(29,41)(35,36)(37,39);(7,9)(10,13,17,29)(11,19,22,37)(12,24,25,41)(14,27,30,40)(15,28,16,32)(18,33,34,39)(20,35,23,38)(21,36,31,26);(10,14)(11,20)(12,15)(13,27)(16,25)(17,30)(18,31)(19,35)(21,34)(22,23)(24,28)(26,33)(29,40)(32,41)(36,39)(37,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17458);