Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #120×#525 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,11,15,13,16,18,22,12,14,17,20,19,21,23,24);(1,2)(3,8)(4,6)(5,7)(10,14)(11,17)(16,21)(18,23);(2,6)(5,8)Polycyclic presentation:
2^8|a^2=e,d^2=fh,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=cfh,d^b=dfh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129651, #129652, #129662, … |
| maximal normal subgroup of | #129651, #129652, #129662, … |
| maximal quotient of | #124290, #124328, #124332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4614);