Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| 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 | |
190 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 190 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,15,20)(11,19,14,24)(12,22,21,16)(13,17,23,18);(2,5)(3,8)(6,7)(9,11,15,14)(10,16,20,22)(12,23,21,13)(17,19,18,24);(9,12,15,21)(10,17,20,18)(11,13,14,23)(16,24,22,19)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151194, #151195, #151234, … |
| maximal normal subgroup of | #151194, #151195, #151234, … |
| maximal quotient of | #125883, #125937, #126041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5201);