Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #331 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,18,14,19,11,21,15,20,22,16,13,17,23,24);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,20,18)(12,13,22,14)(17,24,19,21)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=f,d^2=g,e^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94540, #94541, #94594, … |
| maximal normal subgroup of | #94540, #94541, #94594, … |
| maximal quotient of | #92818, #92828, #92830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 96);