Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×#388 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(10,11)(13,14,16,15);(9,10,12,11)(13,15,16,14);(6,8)(9,11,12,10)112 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#388, #354380, #354381, … |
| maximal normal subgroup of | C8×#388, #354380, #354381, … |
| maximal quotient of | C4×#1437, C4×#1436, C4×#1434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6751);