Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#29509 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(22,23,28,32)(24,29,30,36)(25,34,37,33)(26,31,27,35);(6,8,9,13)(7,10,11,12)(14,15)(16,19)(17,18)(20,21)(22,24)(23,29)(25,35)(26,34)(27,33)(28,30)(31,37)(32,36);(15,18)(19,21)(22,25,26,36,28,37,27,29)(23,30,35,33,32,24,31,34);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(22,26,28,27)(23,31,32,35)(24,33,30,34)(25,36,37,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22497);