Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31076 |
| automorphism | Aut(G) = #72340863 (order 4096) · Inn(G) = H(3,𝔽4) · Out(G) = D42 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2804 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,4,9,6,10,5,8)(3,11,12,16,14,15,13,7);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16);(2,8,10,9)(3,12,14,13);(3,12,14,13)(7,15,16,11)Polycyclic presentation:
2^8|a^2=efg,c^2=fg,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125842, #125843, #125958, … |
| maximal normal subgroup of | #125842, #125843, #125958, … |
| maximal quotient of | #386802, #387233, #387390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25692);