Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | Aut(G) = #72231951 (order 2048) · Inn(G) = #2760 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2760 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #2021 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 89 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)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15);(1,3)(2,6)(4,13)(5,11)(7,15)(8,9)(10,14)(12,16);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12);(6,9)(15,16)Polycyclic presentation:
2^8|a^2=fg,c^2=ef,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,f^b=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151342, #151343, #151462, … |
| maximal normal subgroup of | #151342, #151343, #151462, … |
| maximal quotient of | #346224, #346225, #346226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25688);