Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1210 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)(18,22)(19,23)(24,34)(25,32)(26,33)(27,35)(28,36)(29,37)(30,38)(31,40)(39,44)(41,43)(42,45);(6,8,9,13)(7,10,11,12)(14,16)(15,20)(17,24)(18,25)(19,26)(21,34)(22,32)(23,33)(27,39)(28,40)(29,41)(30,42)(31,36)(35,44)(37,43)(38,45);(14,17,19,28)(15,21,23,36)(16,24,26,40)(18,29,30,27)(20,31,33,34)(22,37,38,35)(25,41,42,39)(32,44,45,43);(7,11)(10,12)(14,18,19,30)(15,22,23,38)(16,25,26,42)(17,27,28,29)(20,32,33,45)(21,35,36,37)(24,39,40,41)(31,43,34,44)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16977);