Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12416 |
| 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 | C4×C8 | C23 |
| prime radical | #493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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:
(2,5)(3,6)(7,8)(9,10)(11,19)(12,15)(13,25)(14,28)(16,21)(17,27)(18,24)(20,35)(22,39)(23,36)(26,34)(29,33)(30,37)(31,38)(32,40);(1,2,6,8,4,7,3,5)(10,15)(16,30)(17,31)(18,28)(19,35)(25,36)(32,34)(33,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,33,31,39)(18,34,28,32)(19,36,35,25)(21,38,37,27)Polycyclic presentation:
2^8|b^2=dfh,c^2=h,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,e^c=eg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12429, #490452, #490454, … |
| maximal normal subgroup of | C2×#12429, #490452, #490454, … |
| maximal quotient of | #123376, #123378, #123380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5417);