Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17841 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C2×C5×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,20)(11,14)(12,18)(16,17)(19,21)(22,23,26,30,27,31,25,29)(24,32,35,36,33,37,34,28);(6,8,9,17,11,19,10,18)(7,12,20,21,14,16,13,15)(22,24,26,35,27,33,25,34)(23,28,29,37,31,36,30,32);(6,7)(8,16)(9,20)(10,13)(11,14)(12,19)(15,17)(18,21)(24,33)(28,36)(32,37)(34,35);(6,9,11,10)(7,13,14,20)(8,17,19,18)(12,15,16,21)(22,25,27,26)(23,29,31,30)(24,34,33,35)(28,37,36,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10829);