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×#23914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1706 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)(12,16,21,31)(13,23,24,35)(14,26,29,18)(17,28,33,25)(20,36,37,40)(22,34,27,32)(30,39,41,38);(6,7)(8,9)(10,12,15,21)(11,16,19,31)(13,20,24,37)(14,27,29,22)(17,30,33,41)(18,34,26,32)(23,40,35,36)(25,39,28,38);(7,9)(10,13)(11,17)(12,20)(14,28)(15,24)(16,30)(18,35)(19,33)(21,37)(22,39)(23,26)(25,29)(27,38)(31,41)(32,40)(34,36);(12,21)(14,29)(16,31)(18,26)(20,37)(23,35)(25,28)(30,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16902);