Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | Aut(G) = C22×#6346637 (order 2048) · Inn(G) = #327 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #327 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C2×#97 | C22 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,19,15,22,31,18,16,20,30,37,14,21,12,29)(11,23,25,34,27,39,24,35,28,38,40,17,26,33,32,36);(2,5)(3,8)(6,7)(9,11,16,28)(10,17,20,34)(12,25,31,40)(13,32,30,24)(14,27,15,26)(18,33,29,39)(19,23,37,38)(21,36,22,35)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fh,d^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=def,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7111, #124066, #124164, … |
| maximal normal subgroup of | C2×#7111, #124066, #124164, … |
| maximal quotient of | #92820, #92838, #92842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 99);