Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7538 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | D32 |
| commutator | C32 | C2×C4×C5 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ C2×C5×C32 ◁ G | |
90 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)(6,7,8,9)(11,17)(12,22)(13,24)(14,29)(15,33)(18,40)(19,44)(20,47)(21,39)(23,28)(25,55)(26,59)(27,36)(30,64)(31,41)(32,37)(34,45)(35,38)(42,72)(43,71)(46,73)(48,49)(50,57)(51,65)(52,63)(53,54)(56,61)(58,69)(60,62)(66,70)(67,68);(10,11,16,21)(12,23,27,52)(13,28,31,63)(14,32,34,66)(15,35,33,20)(17,36,39,22)(18,41,43,24)(19,45,46,29)(25,56,58,72)(26,60,57,51)(30,65,53,62)(37,69,70,55)(38,50,47,59)(40,67,71,48)(42,54,61,64)(44,68,73,49)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 526);