Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,18,24,16)(11,15,19,23,21,12,20,22);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,21)(16,22)(17,23)(20,24);(1,4,5,8)(2,6,3,7)(10,16)(11,19)(13,24)(17,18)(20,21)(22,23);(2,3)(6,7)(9,12,14,15)(10,11,18,21)(13,23,24,22)(16,19,17,20)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6669705, #6669707, #6669709, … |
| maximal normal subgroup of | #6669705, #6669707, #6669709, … |
| maximal quotient of | #344209, #344210, #344213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24173);