Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(3,4,5,6);(4,6);(7,8,9,10,11,12,13,14);(8,12)(10,14)100 conjugacy classes — view the full character table.
| maximal subgroup of | #105694, #105699, #105994, … |
| maximal normal subgroup of | #105694, #105699, #105994, … |
| maximal quotient of | C2×M16×D8, C2×M16×SD16, C2×M16×Q16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53126);