Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1642 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)(8,18)(9,14)(10,23)(11,26)(12,17)(13,20)(15,24)(16,28)(19,29)(21,37)(22,30)(25,33)(27,32)(31,36)(34,35);(6,8,12,9)(7,13,17,30)(10,19,25,21)(11,22,28,20)(14,26,18,16)(15,29,32,37)(23,34,33,31)(24,36,27,35);(6,9,12,8)(7,14,17,18)(10,21,25,19)(11,20,28,22)(13,16,30,26)(15,31,32,34)(23,29,33,37)(24,35,27,36);(6,10)(7,15)(8,19)(9,21)(11,27)(12,25)(13,29)(14,31)(16,33)(17,32)(18,34)(20,36)(22,35)(23,26)(24,28)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16863);