Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×D4 ◁ C2×C5×H(5,𝔽2) ◁ G | |
1252 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)(8,16,19,29)(9,13)(10,22,24,14)(11,15)(12,18)(17,30,31,37)(20,27)(21,34,33,28)(23,26)(25,32)(35,36);(6,8,11,19)(7,12,15,26)(9,17,20,31)(10,23,24,18)(13,25,27,36)(14,29,22,16)(21,35,33,32)(28,37,34,30);(6,9)(7,13)(8,17)(10,21)(11,20)(12,25)(14,28)(15,27)(16,30)(18,32)(19,31)(22,34)(23,35)(24,33)(26,36)(29,37);(9,20)(13,27)(17,31)(21,33)(25,36)(28,34)(30,37)(32,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26349);