Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#2127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
755 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 755 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,11,12,18)(6,14,16,9)(7,10)(8,13)(15,17);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,14)(13,16);(5,12)(8,17)(11,18)(13,15)46 conjugacy classes — view the full character table.
| maximal subgroup of | #146654, #146675, #146702, … |
| maximal normal subgroup of | #146654, #146675, #146702, … |
| maximal quotient of | C2×#11432, C2×#11448, C2×#11479, … |
| automorphism group of | #1414 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26343);