Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11,15,20)(12,21,25,17)(13,26,28,35)(14,16,30,19)(18,29,34,27)(22,31,24,33)(23,38,39,41)(32,37,36,40);(6,8,9,7)(11,16)(12,22)(14,30)(18,26)(19,20)(23,37)(24,25)(27,29)(31,33)(34,35)(38,41)(39,40);(7,8)(10,12,15,25)(11,17,20,21)(13,23,28,39)(14,24,30,22)(16,31,19,33)(18,32,34,36)(26,41,35,38)(27,37,29,40);(10,13)(11,18)(12,23)(14,29)(15,28)(16,26)(17,32)(19,35)(20,34)(21,36)(22,37)(24,40)(25,39)(27,30)(31,41)(33,38)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18174);