Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1946 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)(12,22,24,34)(13,26,28,35)(14,20,15,19)(17,29,32,25)(18,30,33,27)(23,31,37,41)(36,38,40,39);(6,7)(8,9)(10,12)(11,17)(13,23)(14,29)(15,25)(16,24)(18,31)(19,34)(20,22)(21,32)(26,40)(27,39)(28,37)(30,38)(33,41)(35,36);(7,9)(10,13)(11,18)(12,23)(14,27)(15,30)(16,28)(17,31)(19,26)(20,35)(21,33)(22,36)(24,37)(25,38)(29,39)(32,41)(34,40);(12,24)(17,32)(22,34)(23,37)(25,29)(31,41)(36,40)(38,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16396);