Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25182 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,17)(13,25)(14,16)(15,20)(18,26)(19,29)(21,32)(22,37)(23,30)(24,33)(27,34)(28,35)(31,36)(38,40)(39,41);(6,8,9,7)(11,16)(12,21)(14,29)(18,25)(19,20)(22,36)(23,24)(26,27)(30,32)(34,35)(37,40)(38,39);(7,8)(10,12,15,24)(11,17,20,33)(13,22,28,38)(14,23,29,21)(16,30,19,32)(18,31,34,41)(25,37,35,40)(26,36,27,39);(10,13)(11,18)(12,22)(14,27)(15,28)(16,25)(17,31)(19,35)(20,34)(21,36)(23,39)(24,38)(26,29)(30,37)(32,40)(33,41)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18170);