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×#13101 |
| 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 | |
630 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)(6,7,12,18,13,19,11,17)(8,20,25,50,26,51,24,49)(9,27,30,15,31,43,29,44)(10,32,21,47,35,16,36,38)(14,37,40,33,41,59,46,23)(22,53,56,48,57,68,55,65)(28,62,52,42,34,61,63,45)(39,58,66,69,67,54,60,64);(6,8,21,37)(7,14,38,50)(9,22,52,64)(10,33,12,24)(11,25,35,23)(13,26,36,59)(15,39,61,53)(16,20,18,40)(17,46,32,51)(19,41,47,49)(27,60,45,48)(28,58,30,55)(29,56,34,54)(31,57,63,69)(42,65,43,66)(44,67,62,68);(6,9,13,31)(7,15,19,44)(8,22,26,57)(10,34,35,28)(11,30,12,29)(14,39,41,67)(16,45,32,42)(17,43,18,27)(20,48,51,65)(21,52,36,63)(23,58,33,54)(24,56,25,55)(37,64,59,69)(38,61,47,62)(40,60,46,66)(49,68,50,53)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6089);