Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32870 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
502 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)(6,7,10,14,12,17,11,16)(8,18,21,13,22,28,19,29)(9,15,25,30,26,31,24,32)(20,33,35,27,23,36,34,37);(6,8,10,21,12,22,11,19)(7,13,16,18,17,29,14,28)(9,23,24,35,26,20,25,34)(15,27,30,36,31,37,32,33);(7,14)(8,19)(9,24)(10,11)(15,31)(16,17)(18,28)(20,23)(21,22)(25,26)(27,36)(33,37);(6,9)(7,15)(8,20)(10,25)(11,24)(12,26)(13,27)(14,30)(16,32)(17,31)(18,33)(19,34)(21,35)(22,23)(28,36)(29,37)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25858);