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×#32785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
686 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)(8,10)(9,11)(12,13)(14,15,18,17)(16,20,21,19)(22,23,26,30)(24,31,33,28)(25,34,27,29)(32,37,35,36);(6,8)(7,10)(9,12)(11,13)(14,16,18,21)(15,19,17,20)(24,32)(25,27)(28,36)(29,34)(31,37)(33,35);(7,11)(10,13)(15,17)(19,20)(22,24,26,33)(23,28,30,31)(25,32,27,35)(29,37,34,36);(8,12)(10,13)(14,17,18,15)(16,20,21,19)(22,25,26,27)(23,29,30,34)(24,32,33,35)(28,37,31,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25773);