Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13096 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1898 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C2×C5×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×2−1+4 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,18,23,47)(9,24,27,43)(10,28,19,15)(11,16)(12,17)(13,21)(14,26,39,33)(20,49)(22,46,32,63)(25,58)(29,37)(30,57)(31,60,48,41)(34,44)(35,64,67,69)(36,54)(38,59)(40,62)(42,61)(45,55)(50,56,53,66)(51,65)(52,68);(6,8,19,21,12,23,10,29)(7,13,28,47,17,37,15,18)(9,20,25,53,27,52,30,50)(11,32,48,61,34,22,31,54)(14,35,38,65,39,67,40,45)(16,42,60,46,44,36,41,63)(24,56,57,49,43,66,58,68)(26,55,59,69,33,51,62,64);(6,9)(7,14)(8,20)(10,30)(11,33)(12,27)(13,35)(15,40)(16,43)(17,39)(18,45)(19,25)(21,53)(22,55)(23,52)(24,44)(26,34)(28,38)(29,50)(31,59)(32,51)(36,56)(37,67)(41,57)(42,66)(46,68)(47,65)(48,62)(49,63)(54,69)(58,60)(61,64)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6084);