Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(1,3)(2,5)(4,8)(6,7);(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(3,8)(5,7);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^8|a^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250915, #250947, #250979, … |
| maximal normal subgroup of | #250915, #250947, #250979, … |
| maximal quotient of | D4×#417, #6379294, #6379712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54150);