Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#32847 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,9,11)(8,10,12,13)(14,15)(16,20)(17,21)(18,19)(22,28)(23,26)(24,27)(25,29)(30,31)(32,36)(33,42)(34,37)(35,38)(39,43)(40,44)(41,45);(6,8)(7,10)(9,12)(11,13)(15,19)(17,24)(20,26)(22,25)(30,32,35,41)(31,36,38,45)(33,40,34,43)(37,39,42,44);(7,11)(10,13)(14,16,18,23)(15,20,19,26)(17,25,24,22)(21,29,27,28)(30,31)(32,39)(33,37)(34,42)(35,38)(36,43)(40,45)(41,44);(8,12)(10,13)(14,17,18,24)(15,21,19,27)(16,22,23,25)(20,28,26,29)(30,33,35,34)(31,37,38,42)(32,40,41,43)(36,39,45,44)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25835);