Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | Aut(G) = C22×D4×B(4,𝔽2) (order 2048) · Inn(G) = #342 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C24 | C42 |
| Frattini, cosocle | C2×C4×D4 | C22 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,6,4,5,7,8)(10,13);(2,5)(3,7)(9,10,11,14,12,13,15,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=h,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7321, #128096, #128097, … |
| maximal normal subgroup of | C2×#7321, #128096, #128097, … |
| maximal quotient of | #93483, #93496, #93502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 309);