Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #121 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1954 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,13,7,12,23,28,8,14,22,10,5,11,4,21)(3,15,19,31,17,27,16,30,20,29,24,32,18,9,25,26);(1,3,4,16,5,18,22,19,8,20,23,25,7,17,6,24)(2,9,21,31,11,29,10,26,14,27,28,32,12,15,13,30);(1,4,5,22,8,23,7,6)(2,10,12,21,14,13,11,28)(3,16,18,19,20,25,17,24)(9,26,15,31,27,30,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136678, #136679, #136682, … |
| maximal normal subgroup of | #136678, #136679, #136682, … |
| maximal quotient of | #140512, #140518, #140540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6185);