Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C16 · Schur multiplier (H2) = C1 |
| direct product | C5×#7511 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C42 |
| commutator | C4 | C4×C5×C16 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C2×C5 | #1397 |
| prime radical | C22×C5 | M64 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7,8,14,9,16,20,35,10,17,21,36,23,39,44,60,11,18,22,33,24,40,45,56,26,41,46,57,47,62,64,68,12,15,19,34,25,37,42,58,27,38,43,59,48,61,63,69,28,13,29,30,49,31,51,53,50,32,52,54,65,55,66,67);(7,13,15,18)(8,19)(14,33,34,30)(16,31,37,40)(17,32,38,41)(20,42)(21,43)(22,29)(35,56,58,53)(36,57,59,54)(39,55,61,62)(44,63)(45,51)(46,52)(60,68,69,67)(64,66)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 499);