Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,8,13,10,14,9,19)(7,15,17,20,18,11,16,12);(6,11,14,15)(7,16)(8,9)(12,13,20,19)(17,18);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136008, #136011, #136061, … |
| maximal normal subgroup of | #136008, #136011, #136061, … |
| maximal quotient of | #135826, #135846, #136009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5917);