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×#32839 |
| 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 | |
578 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)(31,35)(36,44)(37,42)(38,45);(6,8)(7,10)(9,12)(11,13)(15,19)(17,24)(20,26)(22,25)(30,31,33,35)(32,38,39,45)(34,42,41,37)(36,43,44,40);(7,11)(10,13)(14,16,18,23)(15,20,19,26)(17,25,24,22)(21,29,27,28)(30,32,33,39)(31,36,35,44)(34,43,41,40)(37,45,42,38);(8,12)(10,13)(14,17,18,24)(15,21,19,27)(16,22,23,25)(20,28,26,29)(31,35)(32,39)(37,42)(40,43)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25827);