Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12850 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | #432 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
430 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)(6,7)(8,9)(10,11,14,21,15,18,30,34,16,19,29,35,13,20,27,36)(12,22,25,37,26,31,39,40,23,17,28,32,24,33,38,41);(10,12)(11,17)(13,24)(14,28)(15,26)(16,23)(18,33)(19,22)(20,31)(21,37)(25,29)(27,39)(30,38)(32,35)(34,40)(36,41);(7,9)(11,18,19,20)(12,23)(14,29)(17,31,22,33)(21,36,35,34)(24,26)(27,30)(32,41,37,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5838);