Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22⋊C4 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2);(3,4);(5,6,7,8)(9,10)(11,13)(12,15)(14,16);(10,13,15,14)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#339 |
| maximal subgroup of | #123430, #123431, #123436, … |
| maximal normal subgroup of | #123430, #123431, #123436, … |
| maximal quotient of | C22×#1276, C2×#7867, C2×#7869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13329);