Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
173 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,16,15,20,8,7,11,14,19,12,10,13,18,17);(1,2)(3,4)(5,7,15,16,11,17,10,12)(6,9,13,8,14,19,20,18);(2,4)(5,6)(7,9)(8,16)(10,20)(11,14)(12,18)(13,15)(17,19)Polycyclic presentation:
2^8|a^2=dg,b^2=deg,d^2=g,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #126466, #126467, #126468, … |
| maximal normal subgroup of | #126466, #126467, #126468, … |
| maximal quotient of | #146196, #146199, #146204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6521);