Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#24319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C23×C5 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
930 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)(6,7,10,13)(8,12,9,11)(14,15)(16,20)(17,27)(18,30)(19,23)(21,29)(22,33)(24,41)(25,39)(26,36)(28,40)(31,35)(32,38)(34,43)(37,44)(42,45);(6,8,10,9)(7,11,13,12)(14,16,19,26)(15,20,23,36)(17,24,28,42)(18,31,33,25)(21,37,38,34)(22,39,30,35)(27,41,40,45)(29,44,32,43);(14,17,19,28)(15,21,23,38)(16,24,26,42)(18,32,33,29)(20,34,36,37)(22,40,30,27)(25,44,31,43)(35,45,39,41);(6,9,10,8)(7,12,13,11)(17,28)(21,38)(24,42)(27,40)(29,32)(34,37)(41,45)(43,44)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17307);