Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| 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 | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×Q8 | C22 |
| prime radical | C23 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 128 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14,15,21)(16,24,23,22);(2,5)(3,7)(9,10,12,16,13,15,20,23)(11,14,17,22,18,21,19,24)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=f,e^2=h,b^a=bc,c^a=cf,e^a=egh,c^b=cg,d^b=dfh,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7323, #128104, #128105, … |
| maximal normal subgroup of | C2×#7323, #128104, #128105, … |
| maximal quotient of | #93485, #93497, #93503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 311);