Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23);(2,5)(3,8)(6,7)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,22,16,23);(10,15,17,16)(14,22,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,d^b=df,e^b=egh,d^c=dg,e^d=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151210, #151211, #151246, … |
| maximal normal subgroup of | #151210, #151211, #151246, … |
| maximal quotient of | #125921, #125941, #126080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5235);