Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 731 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(5,6)(7,14,16,11)(8,10)(9,13)(12,17)(15,19,20,18);(1,2)(3,4)(6,10)(8,17)(11,18)(12,13)(14,19)(15,20);(2,4)(5,7)(6,11)(8,15)(9,16)(10,18)(12,19)(13,14)(17,20)Polycyclic presentation:
2^8|a^2=g,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133382, #133384, #133386, … |
| maximal normal subgroup of | #133382, #133384, #133386, … |
| maximal quotient of | #111233, #111234, #111551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3972);