Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12617 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1752 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×M32 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,11,17,12,18,33,47,13,19,34,48,10,16,31,45)(8,20,23,49,22,38,52,46,24,44,32,60,30,14,35,37)(9,15,27,41,29,43,59,68,28,42,58,67,26,40,56,65)(21,39,50,63,25,53,54,69,51,64,61,66,55,36,57,62);(6,8)(7,14)(9,25,28,55)(10,30)(11,32)(12,22)(13,24)(15,39,42,64)(16,44)(17,46)(18,20)(19,38)(21,29,51,26)(23,34)(27,57,58,54)(31,52)(33,35)(36,43,53,40)(37,48)(41,66,67,63)(45,49)(47,60)(50,56,61,59)(62,65,69,68);(6,9,11,27,12,29,33,59,13,28,34,58,10,26,31,56)(7,15,17,41,18,43,47,68,19,42,48,67,16,40,45,65)(8,21,23,50,22,25,52,54,24,51,32,61,30,55,35,57)(14,36,37,62,20,39,49,63,38,53,46,69,44,64,60,66)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5605);