Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C22⋊C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1287 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(9,10,11,12)(14,16);(13,14)(15,16);(2,4)(5,6,7,8)(9,12,11,10);(1,2,3,4)100 conjugacy classes — view the full character table.
| maximal subgroup of | #109647, #109648, #111179, … |
| maximal normal subgroup of | #109647, #109648, #111179, … |
| maximal quotient of | C22⋊C4×#95, C4⋊C4×#95, #153728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6788);