Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | M16×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
397 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);(2,6)(4,8);(9,10,11,12,13,14,15,16);(10,16)(11,15)(12,14)70 conjugacy classes — view the full character table.
| maximal subgroup of | #136472, #136474, #137013, … |
| maximal normal subgroup of | #136472, #136474, #137013, … |
| maximal quotient of | M16×#102, M16×#107, #183949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11022);