Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#1277 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
415 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16,17,19,18,20,21,22);(4,6)(7,8,10,13,11,14,9,12)112 conjugacy classes — view the full character table.
| maximal subgroup of | #93388, #93391, #93429, … |
| maximal normal subgroup of | #93388, #93391, #93429, … |
| maximal quotient of | C2×#7015, #94894, #94906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 859);