Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,22)(18,27)(21,36)(23,37)(24,35)(26,31)(28,39)(30,34)(32,40)(33,38);(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,34)(19,33)(20,25)(22,27)(24,38)(28,40)(29,32)(35,36)(37,39);(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,33,32)(18,25,35,39)(20,36,37,34)(22,26,38,40)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,d^b=df,f^b=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #128533, #128534, #128589, … |
| maximal normal subgroup of | #128533, #128534, #128589, … |
| maximal quotient of | #122440, #122446, #122452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4579);