Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,13,16)(8,11,19,12)(9,20,24,17)(10,18,27,32)(14,22,29,23)(15,26,30,25)(21,28,36,37)(31,34,33,35);(6,8,11,16,13,19,12,7)(9,14,22,17,24,29,23,20)(10,18,25,30,27,32,26,15)(21,28,34,31,36,37,35,33);(6,9)(7,14)(8,17)(10,21)(11,22)(12,23)(13,24)(15,28)(16,29)(18,31)(19,20)(25,34)(26,35)(27,36)(30,37)(32,33);(6,10,12,26,13,27,11,25)(7,15,19,32,16,30,8,18)(9,21,23,35,24,36,22,34)(14,28,20,33,29,37,17,31)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17285);