Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,20,24)(16,25,26,42)(17,22,19,34)(18,30,33,44)(21,36,37,28)(23,29,41,32)(27,43,35,40)(31,38,45,39);(6,8,9,13)(7,10,11,12)(16,26)(21,37)(25,42)(27,35)(28,36)(31,45)(38,39)(40,43);(8,13)(10,12)(14,16)(15,21)(17,27)(18,31)(19,35)(20,26)(22,38)(23,40)(24,37)(25,30)(28,32)(29,36)(33,45)(34,39)(41,43)(42,44);(7,11)(10,12)(14,17,20,19)(15,22,24,34)(16,27,26,35)(18,29,33,32)(21,38,37,39)(23,30,41,44)(25,43,42,40)(28,31,36,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26770);