Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10914 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
698 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)(10,11,14,18,15,19,26,32)(12,20,23,34,22,16,27,29)(13,17,24,30,25,31,37,40)(21,33,36,41,35,28,38,39);(10,12)(11,16,19,20)(13,21)(14,23)(15,22)(17,28,31,33)(18,29,32,34)(24,36)(25,35)(26,27)(30,39,40,41)(37,38);(7,9)(10,13,14,24,15,25,26,37)(11,17,18,30,19,31,32,40)(12,21,23,36,22,35,27,38)(16,28,29,39,20,33,34,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3902);