Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
196 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,16)(13,15)(17,18)(19,21)(20,28)(22,27)(23,25)(24,31)(26,32)(29,30);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,18)(20,29,30,28)(22,27,23,25)(24,32,26,31);(10,13)(11,16)(14,15)(17,20,21,30)(18,24,19,26)(22,32,23,31)(25,29,27,28)Polycyclic presentation:
2^8|a^2=dg,b^2=d,c^2=g,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94715, C2×#10361, #237891, … |
| maximal normal subgroup of | #94715, C2×#10361, #237891, … |
| maximal quotient of | #104714, #104715, #104716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3349);