Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33546 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C5 |
| socle | C2×C5 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
1236 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)(9,11)(10,13)(14,17)(15,19)(16,18)(20,21);(7,11)(12,17)(13,18)(19,21);(9,16)(14,20);(6,8,10,15)(7,12,13,19)(9,14,16,20)(11,17,18,21)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26534);