Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | Aut(G) = #72340198 (order 4096) · Inn(G) = #2668 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,10)(4,7)(5,9)(6,13)(8,14)(11,16)(12,15);(1,3)(2,6)(4,12)(5,11)(7,10)(8,16)(9,15)(13,14);(3,11)(12,13);(2,7)(3,12)(4,14)(8,9)(10,16)(11,13)Polycyclic presentation:
2^8|e^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151327, #151329, #151331, … |
| maximal normal subgroup of | #151327, #151329, #151331, … |
| maximal quotient of | #331927, #331929, #331933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23422);