Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33662 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C22×D8 |
| prime radical | C5×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
558 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)(8,11)(9,10)(12,13)(14,15)(16,20)(17,21)(18,31)(19,24)(22,28)(23,29)(25,32)(26,41)(27,35)(30,37)(33,38)(34,39)(36,42)(40,44)(43,45);(7,10)(11,13)(14,16,19,27)(15,20,24,35)(17,25,30,40)(18,26,29,39)(21,32,37,44)(22,33,36,43)(23,34,31,41)(28,38,42,45);(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(25,38)(26,39)(30,42)(31,36)(33,34)(35,44)(40,45)(41,43);(6,8,9,12)(7,11,10,13)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26650);