Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,13)(10,12)(14,15)(16,26)(17,32)(18,35)(19,25)(20,24)(21,29)(22,34)(23,33)(27,40)(28,38)(30,42)(31,39)(36,41)(37,45)(43,44);(6,8,10,9)(7,11,12,13)(14,16,19,30)(15,21,25,39)(17,28,34,45)(18,31,20,29)(22,40,32,43)(23,38,41,37)(24,42,35,26)(27,36,44,33);(6,9,10,8)(7,11,12,13)(14,17)(15,22)(16,27)(18,33)(19,34)(20,36)(21,37)(23,24)(25,32)(26,43)(28,29)(30,44)(31,45)(35,41)(38,39)(40,42);(14,17)(15,23)(16,28)(18,36)(19,34)(20,33)(21,38)(22,24)(25,41)(26,43)(27,29)(30,45)(31,44)(32,35)(37,39)(40,42)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19180);