Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5×C8 · Schur multiplier (H2) = C29 |
| direct product | C2×C5×#3407 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C2 | C24×C5×C8 |
| Frattini, cosocle | C8 | C25×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
2074 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);(3,4,6,11,7,12,16,24,8,10,17,22,18,23,28,32)(5,9,14,20,15,21,27,31,13,19,25,29,26,30,33,34);(3,5)(4,9)(6,14)(7,15)(8,13)(10,19)(11,20)(12,21)(16,27)(17,25)(18,26)(22,29)(23,30)(24,31)(28,33)(32,34);(5,13)(9,19)(14,25)(15,26)(20,29)(21,30)(27,33)(31,34);(4,10)(9,19)(11,22)(12,23)(20,29)(21,30)(24,32)(31,34)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55611);