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) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
109 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 109 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,15)(11,19,20,37)(13,25,23,38)(14,28,24,18)(16,32,30,21)(17,34,31,27)(22,39,29,35)(26,36,40,33);(1,2,3,5,4,6,7,8)(10,15)(16,30)(17,31)(18,28)(19,37)(25,38)(33,36)(35,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,35,31,39)(18,36,28,33)(19,38,37,25)(21,34,32,27)Polycyclic presentation:
2^8|a^2=h,b^2=dh,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×#12423, #490427, #490429, … |
| maximal normal subgroup of | C2×#12423, #490427, #490429, … |
| maximal quotient of | #134938, #138608, #138628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5411);