Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1961 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
434 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)(8,19)(9,25,29,58)(10,31)(11,17)(12,35)(13,18)(14,22)(15,28,43,30)(16,32)(20,44)(21,51,53,67)(23,46)(24,48)(26,59,62,45)(27,63,64,42)(33,36)(34,47)(37,61,56,68)(38,54)(39,49)(40,55)(41,52,66,69)(50,65,60,57);(6,8,20,49,12,23,10,22,13,24,32,54,11,34,33,55)(7,14,36,46,17,39,16,19,18,40,31,47,35,38,44,48)(9,26,60,69,30,53,64,68,29,62,50,52,28,21,27,61)(15,41,57,59,25,56,63,51,43,66,65,45,58,37,42,67);(6,9)(7,15,18,43)(8,21,24,53)(10,27)(11,28)(12,30)(13,29)(14,37)(16,42,44,63)(17,25,35,58)(19,45)(20,50)(22,52,55,69)(23,26,34,62)(31,57,36,65)(32,60)(33,64)(38,66)(39,41)(40,56)(46,51)(47,67)(48,59)(49,68,54,61)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6204);