Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#32731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
782 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)(6,7,9,12)(8,11,13,10)(14,15)(16,18)(17,19)(20,21)(22,23)(24,27)(25,29)(26,28)(30,34)(31,35)(32,36)(33,37);(6,8)(7,10)(9,13)(11,12)(14,16)(15,18)(17,20)(19,21)(24,30)(25,32)(27,34)(29,36)(31,33)(35,37);(6,8)(7,11)(9,13)(10,12)(15,19)(18,21)(22,24,26,31)(23,27,28,35)(25,33,32,30)(29,37,36,34);(8,13)(10,11)(16,20)(18,21)(23,28)(27,35)(29,36)(34,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25719);