Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24927 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,21)(17,29)(18,20)(19,24)(22,30)(23,33)(25,36)(26,41)(27,34)(28,37)(31,38)(32,39)(35,40)(42,44)(43,45);(6,8,9,12)(7,10,11,13)(15,20)(16,25)(18,33)(22,29)(23,24)(26,40)(27,28)(30,31)(34,36)(38,39)(41,44)(42,43);(7,11)(10,13)(14,16,19,28)(15,21,24,37)(17,26,32,42)(18,27,33,25)(20,34,23,36)(22,35,38,45)(29,41,39,44)(30,40,31,43);(14,17)(15,22)(16,26)(18,31)(19,32)(20,29)(21,35)(23,39)(24,38)(25,40)(27,43)(28,42)(30,33)(34,41)(36,44)(37,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17915);