Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#32851 |
| 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 ◁ C43×C5 ◁ G | |
532 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)(7,8)(11,12)(14,15,18,21)(16,22,24,19)(17,20,26,29)(23,28,25,27)(30,31,34,38)(32,40,42,36)(33,37,35,39)(41,45,43,44);(6,7,10,8)(9,11,13,12)(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27)(32,41)(33,35)(36,44)(37,39)(40,45)(42,43);(6,8,10,7)(9,11,13,12)(17,26)(20,29)(23,25)(27,28)(30,32,34,42)(31,36,38,40)(33,41,35,43)(37,44,39,45);(6,9,10,13)(7,11,8,12)(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28)(30,33,34,35)(31,37,38,39)(32,41,42,43)(36,44,40,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25839);