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)(9,10,12,17)(11,21,23,15)(13,16,26,32)(14,19,27,20)(18,34,33,28)(22,30,25,35)(24,39,38,31)(29,37,36,40);(1,3,5,4)(2,6,7,8)(9,11,12,23)(10,15,17,21)(13,24,26,38)(14,22,27,25)(16,31,32,39)(18,29,33,36)(19,35,20,30)(28,37,34,40);(1,4,5,3)(2,6,7,8)(10,16)(11,22)(13,28)(14,27)(15,29)(17,32)(18,20)(19,33)(21,36)(23,25)(24,38)(26,34)(30,39)(31,35)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=dh,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×#13456, #499202, #499203, … |
| maximal normal subgroup of | C2×#13456, #499202, #499203, … |
| maximal quotient of | #145784, #145785, #145813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6444);