Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | |
349 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)(9,10,15,17,14,19,13,18)(11,16,21,23,22,12,20,24);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,22)(15,21)(17,23)(18,24);(1,4,5,8)(2,6,3,7)(10,17)(11,20)(13,15)(18,19)(21,22)(23,24);(2,3)(6,7)(9,12,14,16)(10,11,19,22)(13,24,15,23)(17,20,18,21)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6669693, #6669695, #6669697, … |
| maximal normal subgroup of | #6669693, #6669695, #6669697, … |
| maximal quotient of | #344017, #344018, #344021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24171);