Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #102 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #102 |
| commutator | C2×C8 | C2×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 111 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,7,8,5)(4,6)(10,13,14,15)(17,18,20,23,22,26,19,27,21,25,28,31,29,32,30,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,30)(22,29)(24,26)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=d,b^2=eh,c^2=fh,d^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7121, #124083, #124085, … |
| maximal normal subgroup of | C2×#7121, #124083, #124085, … |
| maximal quotient of | #92870, #92881, #92882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 109);