Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | D4×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle | C23 | C23×C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1006 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)(9,10,11,13,12,14,15,16);(6,8)100 conjugacy classes — view the full character table.
| maximal subgroup of | #99883, #99887, #99891, … |
| maximal normal subgroup of | #99883, #99887, #99891, … |
| maximal quotient of | D4×#323, #219717, D4×#324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15385);