Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C16 | C23 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C5×#525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C16 ◁ G | |
710 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)(6,7)(8,9)(10,11,13,18,14,19,23,31,15,17,24,29,25,30,35,39)(12,16,20,27,21,28,32,38,22,26,33,36,34,37,40,41);(7,9);(10,12,13,20,14,21,23,32,15,22,24,33,25,34,35,40)(11,16,18,27,19,28,31,38,17,26,29,36,30,37,39,41);(11,17)(16,26)(18,29)(19,30)(27,36)(28,37)(31,39)(38,41)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26420);