Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×D4 ◁ C2×C5×H(5,𝔽2) ◁ G | |
1240 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)(9,13)(10,22,23,14)(11,15)(12,18,25,32)(17,28)(19,29)(20,26)(21,34,33,27)(24,31,35,37)(30,36);(6,8,11,19)(7,12,15,25)(9,17,20,30)(10,18,23,32)(13,24,26,35)(14,16,22,29)(21,31,33,37)(27,28,34,36);(6,9)(7,13)(8,17)(10,21)(11,20)(12,24)(14,27)(15,26)(16,28)(18,31)(19,30)(22,34)(23,33)(25,35)(29,36)(32,37);(9,20)(13,26)(17,30)(21,33)(24,35)(27,34)(28,36)(31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26338);