Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,12,14,26)(9,21)(13,25,28,36)(15,27,16,22)(18,32)(23,35)(29,37,30,33)(31,34);(6,7)(8,13)(9,22)(10,15)(11,16)(12,23)(14,24)(17,28)(18,33)(19,29)(20,30)(21,26)(25,34)(27,35)(31,37)(32,36);(6,8,10,19)(7,13,15,29)(9,18,21,32)(11,20,24,17)(12,25,27,37)(14,28,16,30)(22,33,26,36)(23,34,35,31);(8,17)(13,28)(18,31)(19,20)(25,36)(29,30)(32,34)(33,37)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10603);