Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8)(9,10)(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(8,10)(11,13,14,17)(12,16,15,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #111558, #129325, #146662, … |
| maximal normal subgroup of | #111558, #129325, #146662, … |
| maximal quotient of | C2×C4×#386, C2×C4×#388, C2×C4×#385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27017);