Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D4×#123 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,8);(5,6)(7,8);(2,4)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16);(1,2)(3,4)(9,11,12,15)(10,14,13,16)70 conjugacy classes — view the full character table.
| maximal subgroup of | #130743, #248467, #248468, … |
| maximal normal subgroup of | #130743, #248467, #248468, … |
| maximal quotient of | D4×#386, D4×#388, D4×#385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29083);