Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,23)(16,25,29,40)(17,22,20,32)(19,35,34,42)(21,36,26,30)(24,33,38,27)(28,39,31,45)(37,43,41,44);(7,10)(11,13)(14,15)(16,26)(17,32)(18,23)(19,24)(20,22)(21,29)(25,36)(27,35)(28,37)(30,40)(31,41)(33,42)(34,38)(39,44)(43,45);(14,16,27,43,20,31,34,36,18,29,33,44,17,28,19,30)(15,21,35,39,22,37,38,40,23,26,42,45,32,41,24,25);(6,8,9,12)(7,11,10,13)(14,17,18,20)(15,22,23,32)(16,28,29,31)(19,33,34,27)(21,37,26,41)(24,35,38,42)(25,39,40,45)(30,44,36,43)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26733);