Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#29821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
800 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)(8,11)(9,10)(12,13)(14,15,19,24,18,23,20,25)(16,26,29,41,28,21,35,37)(17,30,33,39,32,40,34,22)(27,43,31,36,44,38,45,42);(7,10)(11,13)(14,16)(15,21)(17,31)(18,28)(19,29)(20,35)(22,38)(23,26)(24,37)(25,41)(27,34)(30,42)(32,45)(33,44)(36,40)(39,43);(8,12)(11,13)(14,17)(15,22)(16,27)(18,32)(19,34)(20,33)(21,36)(23,39)(24,40)(25,30)(26,42)(28,44)(29,45)(31,35)(37,43)(38,41);(6,8,9,12)(7,11,10,13)(17,32)(22,39)(27,44)(30,40)(31,45)(33,34)(36,42)(38,43)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22809);