Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#26207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
780 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)(9,11)(10,13)(14,15)(16,20)(17,18)(19,21)(22,23,27,26)(24,28,31,34)(25,29,33,35)(30,36,37,32);(6,8,9,13)(7,10,11,12)(15,18)(19,20)(22,24)(23,28)(25,32)(26,34)(27,31)(29,30)(33,36)(35,37);(14,16,17,21)(15,19,18,20)(22,25)(23,29)(24,30)(26,35)(27,33)(28,36)(31,37)(32,34);(7,11)(10,12)(15,18)(19,20)(22,23,27,26)(24,28,31,34)(25,29,33,35)(30,36,37,32)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19195);