Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#12915 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1850 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
374 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)(6,7,10,15,11,16,25,35,12,17,26,36,27,37,48,58)(8,13,20,30,21,31,44,54,19,29,42,52,43,53,63,67)(9,14,23,33,24,34,47,57,22,32,45,55,46,56,64,68)(18,38,40,59,41,60,62,69,39,28,49,50,61,51,65,66);(6,8,10,20,11,21,25,44,12,19,26,42,27,43,48,63)(7,13,15,30,16,31,35,54,17,29,36,52,37,53,58,67)(9,18,23,40,24,41,47,62,22,39,45,49,46,61,64,65)(14,28,33,50,34,51,57,66,32,38,55,59,56,60,68,69);(6,9)(7,14)(8,18)(10,23)(11,24)(12,22)(13,28)(15,33)(16,34)(17,32)(19,39)(20,40)(21,41)(25,47)(26,45)(27,46)(29,38)(30,50)(31,51)(35,57)(36,55)(37,56)(42,49)(43,61)(44,62)(48,64)(52,59)(53,60)(54,66)(58,68)(63,65)(67,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5903);