Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7884 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | C2×D4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C22×C42×C5 ◁ G | |
572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,17)(10,13)(11,16)(12,18)(15,20)(19,21)(22,23,25,29,27,31,26,30)(24,28,33,36,32,35,34,37);(6,8)(7,11,13,19)(9,16)(10,15)(12,14,17,20)(18,21)(22,24,26,34,27,32,25,33)(23,28,30,37,31,35,29,36);(8,15)(11,19)(14,20)(16,21)(24,32)(28,35)(33,34)(36,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 872);