Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C43 |
| direct product | C5×#32817 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
746 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)(6,7,9,10)(8,11,12,13)(14,15,17,19)(16,18,20,21)(22,23,25,27)(24,30,32,28)(26,29,33,35)(31,36,34,37);(7,10)(8,12)(14,16)(15,18)(17,20)(19,21)(23,27)(24,31)(26,33)(28,36)(30,37)(32,34);(8,12)(11,13)(15,19)(18,21)(22,24)(23,28)(25,32)(26,34)(27,30)(29,36)(31,33)(35,37);(6,8,9,12)(7,11,10,13)(16,20)(18,21)(23,27)(24,32)(29,35)(31,34)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25805);