Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22889 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×C22⋊C8 ◁ G | |
1020 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)(8,9)(10,11,15,19,16,20,14,28)(12,21,23,17,24,32,29,33)(13,18,27,37,26,35,30,36)(22,34,25,31,38,41,39,40);(10,12,15,23,16,24,14,29)(11,17,19,32,20,33,28,21)(13,25,27,38,26,39,30,22)(18,34,37,31,35,41,36,40);(7,9)(13,26)(18,35)(22,38)(25,39)(27,30)(31,40)(34,41)(36,37);(10,13)(11,18)(12,22)(14,30)(15,27)(16,26)(17,31)(19,37)(20,35)(21,34)(23,25)(24,38)(28,36)(29,39)(32,41)(33,40)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15877);