Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,18)(9,14)(11,19)(12,17)(13,22)(15,27)(16,25)(20,29)(21,32)(23,30)(24,34)(26,33)(28,31)(35,36);(5,7,10,6,11,14,26,16,12,15,27,17,25,33,9,19)(8,20,22,34,23,35,36,30,24,13,29,18,28,32,21,31);(2,4)(5,8,12,24)(6,13,17,20)(7,18,15,34)(9,22,26,29)(10,21,27,36)(11,28,25,23)(14,30,33,31)(16,35,19,32)Polycyclic presentation:
2^8|a^2=de,b^2=dg,c^2=h,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,f^b=fh,d^c=dg,e^c=egh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13479, #499337, #499338, … |
| maximal normal subgroup of | C2×#13479, #499337, #499338, … |
| maximal quotient of | #145901, #145903, #145910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6467);