Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#26028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
706 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)(8,12)(9,10)(11,13)(14,15,17,19)(16,20,21,18)(22,23)(24,30)(25,28)(26,29)(27,32)(31,36)(33,35)(34,37);(7,10)(11,12)(22,24)(23,27)(25,32)(26,31)(28,30)(29,35)(33,34)(36,37);(14,16,17,21)(15,18,19,20)(24,31)(26,34)(27,35)(29,37)(30,36)(32,33);(6,8,9,13)(7,11,10,12)(23,28)(27,30)(29,37)(35,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19016);