Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C23×#23900 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
279 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 85 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 — 3 generators on 16 points:
(5,8)(9,10,13,11,12,14,16,15);(1,2)(3,5)(4,6)(7,8)(9,11,13,14,12,15,16,10);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,f^2=h,b^a=bd,c^a=ce,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #135938, #135939, #135940, … |
| maximal normal subgroup of | #135938, #135939, #135940, … |
| maximal quotient of | #99313, #99314, #99315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1839);