Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23916 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1694 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)(10,11,15,19)(12,20,22,34)(13,17,24,32)(14,26,27,18)(16,28,31,23)(21,37,38,36)(25,35,29,33)(30,40,41,39);(6,7)(8,9)(13,24)(14,27)(17,32)(18,26)(20,34)(21,38)(23,28)(30,41);(7,9)(10,12)(11,16)(13,21)(14,28)(15,22)(17,30)(18,34)(19,31)(20,26)(23,27)(24,38)(25,40)(29,39)(32,41)(33,36)(35,37);(10,13,15,24)(11,17,19,32)(12,21,22,38)(14,29,27,25)(16,30,31,41)(18,35,26,33)(20,36,34,37)(23,40,28,39)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16904);