Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17623 |
| 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 | |
1014 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)(7,13,16,30)(8,19)(10,23)(14,32,33,35)(15,29,28,34)(17,31,18,26)(20,36)(22,24);(6,7)(8,14)(9,15)(10,26)(11,17)(12,18)(13,20)(16,27)(19,29)(21,35)(22,34)(23,30)(24,33)(25,32)(28,37)(31,36);(6,8)(7,14)(9,23)(10,21)(11,19)(12,22)(13,28)(15,30)(16,33)(17,29)(18,34)(20,37)(24,27)(25,36)(26,35)(31,32);(6,9,11,21)(7,15,17,35)(8,20,19,36)(10,24,23,22)(12,25,27,37)(13,29,31,14)(16,28,18,32)(26,33,30,34)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10611);