Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11084 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #399 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
878 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)(7,9)(10,11,15,20,16,21,14,19)(12,17,25,33,23,31,24,32)(13,18,28,36,27,35,29,37)(22,38,26,41,39,30,40,34);(6,7,8,9)(10,12)(11,17)(13,26,27,40)(14,24)(15,25)(16,23)(18,34,35,41)(19,32)(20,33)(21,31)(22,28,39,29)(30,36,38,37);(10,13)(11,18)(12,22)(14,29)(15,28)(16,27)(17,30)(19,37)(20,36)(21,35)(23,39)(24,40)(25,26)(31,38)(32,41)(33,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4072);