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 | |
936 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)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24);(1,4,5,2)(3,7,8,6)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24);(3,8)(6,7)(11,19)(14,18)(20,24)(22,23);(10,16)(14,18)(15,17)(22,23)Polycyclic presentation:
2^8|a^2=f,c^2=fh,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 | #250931, #250963, #250995, … |
| maximal normal subgroup of | #250931, #250963, #250995, … |
| maximal quotient of | #6379294, #6379298, #6379870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54242);