Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1545 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 1545 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9,11,7)(4,6)(5,8)(10,15,16,13)(12,14);(2,6)(3,10)(8,14)(11,16);(1,3)(2,7)(4,10)(5,11)(6,13)(8,9)(12,16)(14,15);(2,6)(7,13)(8,14)(9,15)Polycyclic presentation:
2^8|a^2=g,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #105752, #107194, #110086, … |
| maximal normal subgroup of | #105752, #107194, #110086, … |
| maximal quotient of | #213492, #213494, #213497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14734);