Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2032 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #393 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,8)(6,13,15,9)(7,10)(11,18,14,16)(12,17);(3,5)(4,8)(6,14,15,11)(7,12)(9,16,13,18)(10,17);(3,6)(4,9)(5,11)(7,15)(8,16)(10,13)(12,14)(17,18)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16461);