Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#32896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
3610 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)(6,7,12,18)(8,19,22,32)(9,15,24,31)(10,17,11,16)(13,25,29,21)(14,26,30,23)(20,27,37,34)(28,36,33,35);(6,8)(7,13)(9,20)(10,25)(11,21)(12,22)(14,27)(15,28)(16,32)(17,19)(18,29)(23,36)(24,37)(26,35)(30,34)(31,33);(6,9)(7,14)(8,20)(10,23)(11,26)(12,24)(13,27)(15,16)(17,31)(18,30)(19,33)(21,35)(22,37)(25,36)(28,32)(29,34);(6,9)(7,15)(8,20)(10,26)(11,23)(12,24)(13,28)(14,16)(17,30)(18,31)(19,34)(21,36)(22,37)(25,35)(27,32)(29,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25884);