Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12917 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
384 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)(10,11,16,22,17,21,34,39,14,20,30,23,15,31,33,40)(12,13,26,29,27,28,19,37,24,18,35,36,25,32,38,41);(7,9)(10,12,16,26,17,27,34,19,14,24,30,35,15,25,33,38)(11,18,22,36,21,32,39,41,20,13,23,29,31,28,40,37);(10,13)(11,19)(12,23)(14,18)(15,32)(16,29)(17,28)(20,38)(21,35)(22,24)(25,39)(26,31)(27,40)(30,36)(33,41)(34,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5905);