Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#121 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
392 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)(3,6)(4,5)(7,8)(9,10,14,16,13,18,12,17)(11,15,20,23,21,24,19,22);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,22)(18,24);(10,16)(11,19)(12,14)(17,18)(20,21)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=fh,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #211076, #211079, #211084, … |
| maximal normal subgroup of | #211076, #211079, #211084, … |
| maximal quotient of | #243296, #243297, #243298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16706);