Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C2×C5×#3514 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
8006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,10)(6,9)(7,8)(12,15)(16,18);(4,8)(9,10);(3,5,7,6)(4,9,8,10)(11,12)(13,16)(14,15)(17,18);(3,4)(5,9)(6,10)(7,8);(3,6,7,5)(4,10,8,9)(11,13,14,17)(12,16,15,18)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55766);