Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#29589 |
| 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 | |
596 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)(6,7,9,11)(8,12,13,10)(22,23,26,32,27,31,36,30)(24,33,35,29,25,28,34,37);(6,8,9,13)(7,10,11,12)(14,15,17,18)(16,19,20,21)(22,24)(23,28)(25,27)(26,35)(29,30)(31,33)(32,37)(34,36);(15,18)(19,21)(22,25,27,24)(23,29,31,37)(26,35,36,34)(28,32,33,30);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(22,26,27,36)(23,30,31,32)(24,34,25,35)(28,37,33,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22577);