Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C22×C4×B(4,𝔽2) (order 1024) · Inn(G) = #99 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C23 | C2×C16 |
| Frattini, cosocle | C2×C22⋊C8 | C22 |
| socle | C22 | #324 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
112 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,5,6,7)(3,8)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cg,d^b=de,d^c=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7064, #122313, #122314, … |
| maximal normal subgroup of | C2×#7064, #122313, #122314, … |
| maximal quotient of | #92871, #92872, #92965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52);