Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#26878 |
| 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 ◁ C4×C5×C4⋊C4 ◁ G | |
614 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)(6,7,10,12)(8,13,9,11)(14,15)(16,19)(17,18)(20,21)(22,23,26,29)(24,31,34,28)(25,35,27,30)(32,36,33,37);(6,8,10,9)(7,11,12,13)(22,24)(23,28)(25,33)(26,34)(27,32)(29,31)(30,37)(35,36);(6,9,10,8)(7,11,12,13)(15,18)(19,21)(23,29)(24,32)(25,27)(28,36)(31,37)(33,34);(7,12)(11,13)(14,16,17,20)(15,19,18,21)(22,25,26,27)(23,30,29,35)(24,33,34,32)(28,37,31,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19866);