Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #72313686 (order 4096) · Inn(G) = #481 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1874 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,22,28,7,9,4,17,19,26,20,30)(3,12,15,25,16,27,21,24,14,31,13,29,23,32,18,8);(1,3)(2,8,9,24)(4,18)(5,21)(6,23)(7,14)(10,27,17,32)(11,29,26,25)(12,30,31,28)(13,20)(15,22)(16,19);(2,9)(4,5)(8,24)(11,26)(13,15)(18,21)(20,22)(25,29)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131314, #131321, #131350, … |
| maximal normal subgroup of | #131314, #131321, #131350, … |
| maximal quotient of | #136617, #136618, #136645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5996);