Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,14,16,21,23,15,20,22,24,12,18,11,17);(2,5)(6,8)(10,16,20,18)(11,21)(13,22)(17,24,23,19);(1,3,4,7)(2,6,5,8)(9,11,12,22,15,21,14,13)(10,17,18,24,20,23,16,19)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=ch,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93440, #94001, #124205, … |
| maximal normal subgroup of | #93440, #94001, #124205, … |
| maximal quotient of | #128208, #128211, #128217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5582);