Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C5 |
| prime radical | C5×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8 ◁ C5×C16 ◁ C5×C32 ◁ C5×C64 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(7,69)(8,132)(9,67)(10,130)(11,65)(12,128)(13,63)(14,126)(15,61)(16,124)(17,59)(18,122)(19,57)(20,120)(21,55)(22,118)(23,53)(24,116)(25,51)(26,114)(27,49)(28,112)(29,47)(30,110)(31,45)(32,108)(33,43)(34,106)(35,41)(36,104)(37,39)(38,102)(40,100)(42,98)(44,96)(46,94)(48,92)(50,90)(52,88)(54,86)(56,84)(58,82)(60,80)(62,78)(64,76)(66,74)(68,72)(71,133)(73,131)(75,129)(77,127)(79,125)(81,123)(83,121)(85,119)(87,117)(89,115)(91,113)(93,111)(95,109)(97,107)(99,105)(101,103)335 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 540);