Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
714 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)(8,10)(9,12)(11,13)(14,15,19,23)(16,24,28,38)(17,21,29,35)(18,31,25,22)(20,32,34,27)(26,37,40,45)(30,42,39,36)(33,43,44,41);(14,16,25,32,19,28,18,27)(15,20,31,38,23,34,22,24)(17,26,39,43,29,40,30,41)(21,33,42,45,35,44,36,37);(7,10)(8,13)(11,12)(14,17)(15,21)(16,26)(18,30)(19,29)(20,33)(22,36)(23,35)(24,37)(25,39)(27,41)(28,40)(31,42)(32,43)(34,44)(38,45);(17,29)(21,35)(26,40)(30,39)(33,44)(36,42)(37,45)(41,43)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12176);