Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3529 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
948 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)(7,10)(8,12)(11,13)(14,15,20,25)(16,26,30,36)(17,31,33,39)(18,24,19,23)(21,28,37,29)(22,34,38,32)(27,35,43,45)(40,42,44,41);(8,12)(10,13)(14,16)(15,21)(17,27)(18,28)(19,29)(20,30)(22,35)(23,36)(24,26)(25,37)(31,44)(32,41)(33,43)(34,42)(38,45)(39,40);(6,7)(8,13)(9,11)(10,12)(14,17)(15,22)(16,27)(18,32)(19,34)(20,33)(21,35)(23,31)(24,39)(25,38)(26,40)(28,41)(29,42)(30,43)(36,44)(37,45);(14,18,20,19)(15,23,25,24)(16,28,30,29)(17,32,33,34)(21,36,37,26)(22,31,38,39)(27,41,43,42)(35,44,45,40)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11984);