Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C2×C5×#3020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
4710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4)(5,13)(6,10)(7,19)(8,11)(9,15)(12,22)(14,30)(16,27)(17,28)(18,25)(20,24)(21,26)(23,31)(29,32)(33,34);(3,5)(4,9)(6,14)(7,20)(8,16)(10,23)(11,24)(12,27)(13,19)(15,22)(17,33)(18,32)(21,31)(25,34)(26,29)(28,30);(3,6,8,18)(4,10,11,25)(5,14,16,32)(7,21,22,17)(9,23,24,34)(12,28,19,26)(13,29,27,30)(15,33,20,31);(4,11)(9,24)(10,25)(12,19)(13,27)(23,34)(26,28)(29,30)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53372);