Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33765 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
616 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)(8,12)(9,11)(10,13)(14,15)(16,20)(17,21)(18,31)(19,24)(22,28)(23,29)(25,33)(26,32)(27,41)(30,37)(34,40)(35,39)(36,42)(38,43)(44,45);(6,8)(7,10)(9,13)(11,12)(14,16,19,25)(15,20,24,33)(17,26,30,38)(18,27,29,39)(21,32,37,43)(22,34,36,44)(23,35,31,41)(28,40,42,45);(8,13)(10,12)(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(26,40)(27,39)(30,42)(31,36)(33,43)(34,35)(38,45)(41,44);(7,11)(10,12)(16,25)(20,33)(26,38)(27,39)(32,43)(34,44)(35,41)(40,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26753);