Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33558 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
1158 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)(10,14)(11,15)(12,19)(17,29)(18,30)(20,26)(21,27)(22,35)(23,28)(24,31)(25,33)(32,37)(34,36);(6,8,11,18)(7,12,15,25)(9,17,20,32)(10,19,23,33)(13,24,26,36)(14,16,28,30)(21,31,22,34)(27,29,35,37);(6,9)(7,13)(8,17)(10,22)(11,20)(12,24)(14,27)(15,26)(16,29)(18,32)(19,34)(21,23)(25,36)(28,35)(30,37)(31,33);(8,18)(9,20)(12,25)(13,26)(16,30)(19,33)(21,22)(27,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26546);