Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(5,6)(7,10,12,9)(8,11);(6,9)(7,12)(10,11);(2,4)(7,12)Polycyclic presentation:
2^8|a^2=f,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152554, #152580, #152581, … |
| maximal normal subgroup of | #152554, #152580, #152581, … |
| maximal quotient of | #129607, #129641, #130092, … |
| outer automorphism group of | C2×#330, #1526, #2986 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5747);