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×#33561 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1096 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)(10,15)(11,16)(12,17)(13,20)(19,30)(21,28)(22,27)(23,31)(24,32)(25,33)(26,34)(29,35)(36,37);(6,8,12,9)(7,13,17,27)(10,21,24,19)(11,20,25,22)(14,16,18,33)(15,29,32,26)(23,35,36,34)(28,37,30,31);(6,9,12,8)(7,14,17,18)(10,19,24,21)(11,20,25,22)(13,16,27,33)(15,30,32,28)(23,35,36,34)(26,37,29,31);(6,10,12,24)(7,15,17,32)(8,19,9,21)(11,23,25,36)(13,26,27,29)(14,28,18,30)(16,31,33,37)(20,34,22,35)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26549);