Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1016 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)(12,23)(13,19)(14,21)(15,20)(16,22)(17,26)(18,28)(24,32)(25,39)(27,38)(29,37)(30,34)(31,35)(33,41)(36,40);(6,8,9,7)(10,12)(11,17)(13,24)(14,30)(15,26)(16,27)(18,32)(19,36)(20,38)(21,23)(22,34)(25,28)(29,41)(31,40)(33,35)(37,39);(10,13,16,29)(11,18,22,35)(12,24,27,41)(14,28,15,31)(17,32,34,33)(19,21,37,20)(23,39,38,36)(25,26,40,30);(10,13)(11,19)(12,25)(14,31)(15,28)(16,29)(17,33)(18,20)(21,35)(22,37)(23,39)(24,30)(26,41)(27,40)(32,34)(36,38)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15884);