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) = C23 |
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 | |
179 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,26)(13,18)(14,31)(15,19)(16,22)(17,29)(21,36)(23,33)(24,40)(25,38)(27,35)(28,34)(30,37)(32,39);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,17)(13,31)(14,18)(15,19)(16,23)(20,36)(22,33)(24,32)(25,35)(26,29)(27,38)(28,37)(30,34)(39,40);(9,11,15,25)(10,16,19,32)(12,27,30,36)(13,24,14,23)(17,33,34,40)(18,35,31,21)(20,37,38,26)(22,28,39,29)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=h,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×#13370, #498971, #498972, … |
| maximal normal subgroup of | C2×#13370, #498971, #498972, … |
| maximal quotient of | #144449, #144451, #144480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6358);