Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1763 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups; only the 391 normal subgroups are shown.
This group has 843 conjugacy classes of subgroups — too many to draw. Only its 391 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,10,9)(11,12)(13,14)(15,17)(16,18);(4,6)(11,13,15,18,16,17,14,12);(7,9,10,8)(11,14,16,15)(12,13,18,17)88 conjugacy classes — view the full character table.
| maximal subgroup of | #106431, #106433, #106917, … |
| maximal normal subgroup of | #106431, #106433, #106917, … |
| maximal quotient of | C2×#8171, C2×#8178, C2×#8173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13398);