Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12754 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C5×#3533 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,13)(10,22,23,14)(11,15)(12,18,25,34)(17,31)(19,30)(20,28,35,26)(21,27)(24,32,36,37)(29,33);(6,8)(7,12)(9,17)(10,18)(11,19)(13,24)(14,16)(15,25)(20,32)(21,33)(22,30)(23,34)(26,31)(27,36)(28,29)(35,37);(6,9,11,21)(7,13,15,27)(8,17,19,33)(10,20,23,35)(12,24,25,36)(14,28,22,26)(16,29,30,31)(18,32,34,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5742);