Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#10552 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,24)(17,20)(18,22)(19,23)(21,27)(25,28)(26,29);(6,8,11,12)(7,9,13,10)(15,20)(16,25)(17,18)(21,24)(22,23)(28,29);(7,8)(9,10)(12,13)(14,16,17,26,19,25,18,27)(15,21,22,28,23,29,20,24)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3540);