Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#19604 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#127 |
| prime radical | C23×C5 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
706 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)(10,19)(11,14)(12,17)(16,20)(18,21)(22,23,26,29)(24,30,33,37)(25,34,31,28)(27,35,36,32);(6,8,10,17,11,18,9,16)(7,12,13,15,14,20,19,21);(22,24,31,35,26,33,25,32)(23,27,34,37,29,36,28,30);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12592);