Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#1736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,9,10)(12,16)(13,14)(17,18);(4,6)(11,12,14,18,15,17,13,16);(12,17)(16,18)88 conjugacy classes — view the full character table.
| maximal subgroup of | #123469, #123473, #123482, … |
| maximal normal subgroup of | #123469, #123473, #123482, … |
| maximal quotient of | C2×#7873, C2×#7876, C2×#7879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13334);