Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| 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 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,18)(14,20)(17,25)(19,26)(21,34)(22,31)(23,29)(24,30)(27,33)(28,35)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,22)(13,21)(14,24)(16,31)(17,28)(18,34)(19,29)(20,30)(23,26)(25,35)(27,36)(32,39)(33,40)(37,38);(1,4,5,3)(2,6,7,8)(10,16)(11,21)(12,25)(13,26)(15,28)(17,19)(18,32)(20,33)(22,36)(23,24)(27,38)(29,40)(30,39)(31,34)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13455, #499202, #499203, … |
| maximal normal subgroup of | C2×#13455, #499202, #499203, … |
| maximal quotient of | #145783, #145785, #145813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6443);