Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
672 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)(14,15,17,18)(16,19,21,20)(22,23,27,30)(24,31,25,28)(26,29,34,37)(32,36,33,35);(6,8)(7,10)(9,12)(11,13)(14,16)(15,19)(17,21)(18,20)(22,24,27,25)(23,28,30,31)(26,33,34,32)(29,36,37,35);(7,11)(10,13)(15,18)(16,20)(19,21)(22,25,27,24)(23,28,30,31)(26,32,34,33)(29,36,37,35);(8,12)(10,13)(16,21)(19,20)(22,26,27,34)(23,29,30,37)(24,32,25,33)(28,35,31,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25775);