Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#2134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×Q8 | C24×C5 |
| prime radical | C2×C5×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
1390 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);(3,4,7,12,8,13,5,10)(6,9,14,17,15,11,16,18);(4,9)(5,7)(6,15)(10,18)(11,13)(12,17);(3,5,8,7)(4,10,13,12)(6,14,15,16)(9,17,11,18)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26355);