Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
538 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)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23)(24,32)(25,30)(26,29)(27,28)(31,35)(33,36)(34,37);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24,25,27)(23,28,30,32)(26,33,34,35)(29,31,37,36);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(24,27)(26,35)(28,36)(30,37)(31,32)(33,34);(7,13)(11,14)(12,21)(19,20)(23,30)(28,32)(29,37)(31,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26781);