Finite Group DatabasebetaFinite Group Databasebeta| order | 2560 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#489347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #18524 |
| commutator, Frattini, cosocle | C4×C8 | C24×C5 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×M16 ◁ G | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,5,10)(3,6,13,16)(4,8,14,7)(9,12,15,11)(17,18,19,20,21);(2,6)(3,11)(4,14)(7,15)(8,9)(10,16)(12,13);(2,7)(3,12)(4,14)(8,10)(9,15)(11,13);(3,11)(4,14)(6,9)(7,8)(12,13)(15,16)220 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,10)(3,6,13,16)(4,8,14,7)(9,12,15,11)(17,18,19,20,21), (2,6)(3,11)(4,14)(7,15)(8,9)(10,16)(12,13), (2,7)(3,12)(4,14)(8,10)(9,15)(11,13), (3,11)(4,14)(6,9)(7,8)(12,13)(15,16));