Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24986 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2420 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
642 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)(8,19)(9,23)(10,14)(11,28)(12,17)(13,18)(15,26)(16,22)(20,29)(21,30)(24,31)(25,33)(27,32)(34,35)(36,37);(6,8,11,10,13,22,12,21)(7,14,17,19,18,30,28,16)(9,24,26,20,27,35,25,36)(15,31,23,37,33,34,32,29);(6,9)(7,15)(8,20)(10,24)(11,25)(12,26)(13,27)(14,29)(16,31)(17,32)(18,33)(19,34)(21,35)(22,36)(23,28)(30,37);(6,10,11,22,13,21,12,8)(7,16,17,14,18,19,28,30)(9,24,25,36,27,35,26,20)(15,31,32,29,33,34,23,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17974);