Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #1964 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1964 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,18,21,19,29,17,32)(4,20,9,27,23,16,22,10)(5,11,15,28,24,31,26,30);(1,3,15,10)(2,9,28,21)(4,11,29,12)(5,16,7,18)(6,17,24,20)(8,19,26,27)(13,22,30,32)(14,25,23,31);(1,4,6,22,8,23,7,9)(2,10,12,20,13,27,25,16)(3,11,17,30,19,31,18,28)(5,21,15,29,24,32,26,14)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fg,g^b=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13232, #495202, #495203, … |
| maximal normal subgroup of | C2×#13232, #495202, #495203, … |
| maximal quotient of | #140615, #140633, #140645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6220);