Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(2,4)(5,6,11,15,10,20,8,18,13,16,9,14,12,7,17,19);(1,2,3,4)(5,7,13,20)(6,10,16,12)(8,18,17,19)(9,15,11,14);(5,8,12,11,13,17,10,9)(6,14,20,19,16,15,7,18)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=gh,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fg,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136865, #136868, #136872, … |
| maximal normal subgroup of | #136865, #136868, #136872, … |
| maximal quotient of | #140849, #140871, #141023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6249);