Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C43 |
| direct product | C4×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C42 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,13,15,19)(8,11)(9,12)(10,14,17,20)(16,18);(5,7)(6,10,12,17)(8,14)(9,15)(11,13,18,19)(16,20)88 conjugacy classes — view the full character table.
| maximal subgroup of | #93290, #93291, #93292, … |
| maximal normal subgroup of | #93290, #93291, #93292, … |
| maximal quotient of | C4×#1274, #105252, #105253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3685);