Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C4×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1419 conjugacy classes of subgroups; only the 815 normal subgroups are shown.
This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 815 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6);(7,8)(9,14)(10,12)(11,13)(15,16)(17,19)(18,20)(21,22);(7,9,11,10)(8,12,13,14)(15,17,18,21)(16,19,20,22);(7,10,11,9)(8,12,13,14)(16,20)(19,22)112 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C4×#126⋊C3 |
| maximal subgroup of | C4×#1473, #181991, #189456, … |
| maximal normal subgroup of | C4×#1473, #181991, #189456, … |
| maximal quotient of | C2×C4×#388, C2×C4×#387, C2×C4×#383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27021);