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×#26925 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C23×C5 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
714 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)(6,7,9,10)(8,12,13,11)(14,15,19,24,21,17,20,25)(16,26,29,32,31,27,30,22)(18,34,35,33,37,39,36,23)(28,38,42,41,44,45,43,40);(7,10)(11,12)(14,16,21,31)(15,22,17,32)(18,28,37,44)(19,30,20,29)(23,38,33,45)(24,27,25,26)(34,40,39,41)(35,43,36,42);(14,17,21,15)(16,27,31,26)(18,33,37,23)(19,24,20,25)(22,29,32,30)(28,41,44,40)(34,36,39,35)(38,43,45,42);(6,8,9,13)(7,11,10,12)(14,18)(15,23)(16,28)(17,33)(19,36)(20,35)(21,37)(22,38)(24,39)(25,34)(26,40)(27,41)(29,43)(30,42)(31,44)(32,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19913);