Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24288 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
1290 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)(6,7)(8,20)(9,25)(10,28)(11,18)(12,17)(13,19)(14,23)(15,27)(16,29)(21,32)(22,37)(24,36)(26,33)(30,35)(31,34);(6,8,11,23,13,24,12,26)(7,14,17,20,19,33,18,36)(9,21,27,10,22,29,31,30)(15,16,25,28,34,32,37,35);(6,9)(7,15)(8,21)(10,23)(11,27)(12,31)(13,22)(14,32)(16,33)(17,25)(18,37)(19,34)(20,35)(24,29)(26,30)(28,36);(6,10,11,29,13,30,12,21)(7,16,17,28,19,32,18,35)(8,22,23,31,24,9,26,27)(14,25,20,34,33,37,36,15)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17276);