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 | #3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10,14,18)(11,15,13,17)(12,20,21,23)(16,22,24,19);(1,2,4,5)(3,7,8,6)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,23,24,20);(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24);(1,3,4,8)(2,6,5,7);(9,11,14,13)(10,17,18,15)(12,19,21,22)(16,20,24,23)Polycyclic presentation:
2^8|a^2=h,b^2=fh,d^2=f,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 | #250934, #250966, #250998, … |
| maximal normal subgroup of | #250934, #250966, #250998, … |
| maximal quotient of | #6379389, #6379457, #6380338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54312);