Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2002 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1375 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8)(9,11)(10,13)(12,14);(8,11)(9,14)(12,13);(4,6)(9,14)(11,12)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2002, #359063, #375815, … |
| maximal normal subgroup of | C4×#2002, #359063, #375815, … |
| maximal quotient of | C2×#10078, C2×#10081, C2×#10079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16383);