Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#10682 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(15,18)(19,20)(22,23,27,28,26,31,25,30)(24,32,35,29,34,37,33,36);(6,8,9,13)(7,10,11,12)(14,15,16,18)(17,19,21,20)(23,28)(25,27)(29,32)(30,31)(33,35)(36,37);(15,19)(17,21)(18,20)(22,24,26,34)(23,29,31,36)(25,35,27,33)(28,32,30,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3670);