Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C32 · Schur multiplier (H2) = C2 |
| direct product | C5×#7510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C32 | C22 |
| commutator | C2 | C4×C5×C32 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle, prime radical | C22×C5 | C2×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C32 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,3,4,5)(10,11,12,18,13,19,22,36,14,20,23,37,26,39,42,58,15,21,24,38,27,40,43,59,29,41,45,60,48,61,62,71,16,17,25,32,28,33,44,52,30,34,46,53,49,55,63,67,31,35,47,54,50,56,64,68,51,57,65,69,66,70,72,73);(6,7,8,9)(11,17)(18,32)(19,33)(20,34)(21,35)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(58,67)(59,68)(60,69)(61,70)(71,73)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 498);