Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12954 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1864 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
380 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)(6,7,11,16,12,18,36,17)(8,19,23,51,24,39,34,46)(9,25,27,47,28,63,35,62)(10,29,32,15,33,48,57,49)(13,37,40,21,42,30,41,22)(14,31,43,26,45,64,44,65)(20,52,55,69,56,38,60,68)(50,59,67,54,61,58,66,53);(6,8,12,24)(7,13)(9,20,28,56)(10,30,33,37)(11,34,36,23)(14,38)(15,46)(16,41)(17,40)(18,42)(19,29)(21,57,22,32)(25,61)(26,59,65,58)(27,60,35,55)(31,54,64,53)(39,48)(43,69)(44,68)(45,52)(47,67)(49,51)(50,63)(62,66);(6,9,12,28)(7,14,18,45)(8,20,24,56)(10,31,33,64)(11,35,36,27)(13,38,42,52)(15,47,49,62)(16,44,17,43)(19,50,39,61)(21,58,22,59)(23,60,34,55)(25,29,63,48)(26,32,65,57)(30,54,37,53)(40,69,41,68)(46,67,51,66)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5942);