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 | #3475 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,14,16,22)(12,18,20,24)(15,21,17,23);(1,4,5,2)(3,7,8,6)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(3,8)(6,7);(10,16)(14,22)(15,17)(21,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,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 | #6542421, #6542424, #6542433, … |
| maximal normal subgroup of | #6542421, #6542424, #6542433, … |
| maximal quotient of | #6379293, #6379299, #6379869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54241);