Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle, prime radical | C22 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 105 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,3,5,6,8,7)(9,10,13,17)(11,19,20,38)(12,23,26,37)(14,28,25,18)(15,30,31,21)(16,33,35,27)(22,39,29,34)(24,36,40,32);(9,11,13,20)(10,15,17,31)(12,24,26,40)(14,29,25,22)(16,34,35,39)(18,36,28,32)(19,37,38,23)(21,33,30,27);(1,3,4,6,5,7,8,2)(11,20)(15,31)(19,38)(21,30)(22,29)(24,40)(32,36)(34,39)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=eh,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=df,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12076, #483783, #483785, … |
| maximal normal subgroup of | C2×#12076, #483783, #483785, … |
| maximal quotient of | #128707, #128747, #129188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5064);