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×#32733 |
| 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 ◁ C2×C5×#117 ◁ G | |
682 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)(22,23,26,25)(24,30,32,28)(27,29,35,34)(31,33,37,36);(6,8)(7,10)(9,13)(11,12)(14,16)(15,18)(17,20)(19,21)(22,24,26,32)(23,28,25,30)(27,36,35,33)(29,37,34,31);(6,8)(7,11)(9,13)(10,12)(15,19)(18,21)(22,25,26,23)(24,31,32,37)(27,29,35,34)(28,36,30,33);(8,13)(10,11)(16,20)(18,21)(23,25)(24,32)(29,34)(33,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25721);