Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1987 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 166 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:
(1,2,5,11,7,12,21,31,8,13,20,10,6,23,4,19)(3,14,15,26,17,30,32,9,18,27,16,28,24,25,22,29);(1,3)(2,9,13,29)(4,15)(5,22)(6,17)(7,24)(8,18)(10,25,11,30)(12,26,23,28)(14,19,27,31)(16,21)(20,32);(1,4,6,20,8,21,7,5)(2,10,12,19,13,11,23,31)(3,15,17,32,18,16,24,22)(9,25,26,27,29,30,28,14)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=g,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fgh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136858, #136860, #136862, … |
| maximal normal subgroup of | #136858, #136860, #136862, … |
| maximal quotient of | #141847, #141848, #141868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6259);