Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C2×C5×#3565 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
6554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,7,8)(5,11,12,9)(6,14,15,10)(13,16,18,17)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34);(4,8)(5,12)(10,14)(13,18)(20,25)(22,26)(28,31)(30,33);(3,5,7,12)(4,9,8,11)(6,13,15,18)(10,16,14,17)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29);(3,6,7,15)(4,10,8,14)(5,13,12,18)(9,16,11,17)(19,23,24,32)(20,28,25,31)(21,29,27,34)(22,30,26,33)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55729);