Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | D4×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1007 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);(2,4);(5,6,7,8)(9,10)(11,13)(12,15)(14,16);(10,13)(11,16)(14,15)70 conjugacy classes — view the full character table.
| maximal subgroup of | #220728, #220729, #220734, … |
| maximal normal subgroup of | #220728, #220729, #220734, … |
| maximal quotient of | D4×D4⋊C8, #219808, #219820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15401);