Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33552 |
| 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 | |
1156 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)(24,32)(25,33)(31,37)(34,36);(6,8)(7,12)(9,17)(10,19)(11,18)(13,24)(14,16)(15,25)(20,32)(21,31)(22,34)(23,33)(26,29)(27,36)(28,30)(35,37);(6,9,11,21)(7,13,15,27)(8,17,18,31)(10,22,23,20)(12,24,25,36)(14,26,28,35)(16,29,30,37)(19,34,33,32);(8,18)(12,25)(16,30)(17,31)(19,33)(24,36)(29,37)(32,34)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26540);