Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C26 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
5647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(5,6)(7,14)(8,13)(9,17)(10,12)(11,15)(16,20)(18,19);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(6,12)(11,19)(13,17)(14,20);(5,8,10,9)(6,13,12,17)(7,15,16,18)(11,20,19,14)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56071);