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×#33557 |
| 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 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12)(9,13)(10,14)(11,16)(15,20)(17,19)(18,21);(6,8)(7,11)(9,16)(10,15)(12,13)(14,19)(17,18)(20,21);(6,8)(7,12)(9,17)(10,15)(11,13)(14,20)(16,18)(19,21);(8,15)(11,19)(12,20)(16,17)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26545);