Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C32 · Schur multiplier (H2) = C2 |
| direct product | C5×#7513 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C16 | C22⋊C4 |
| commutator | C22 | C2×C5×C32 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C2×C5 | C22⋊C32 |
| prime radical | C23×C5 | C32 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C16 ◁ C2×C5×C32 ◁ G | |
90 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)(6,7,8,14,9,15,19,32,10,16,20,33,22,35,41,54,11,17,21,34,23,36,42,55,25,37,43,56,44,57,61,67,12,13,18,28,24,29,38,48,26,30,39,49,45,51,58,63,27,31,40,50,46,52,59,64,47,53,60,65,62,66,68,69);(7,13)(8,18)(15,29)(16,30)(17,31)(19,38)(20,39)(21,40)(35,51)(36,52)(37,53)(41,58)(42,59)(43,60)(57,66)(61,68)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 501);