Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = C22×#60399 (order 1024) · Inn(G) = B(4,𝔽2) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| socle | C2 | #1867 |
| prime radical | #525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,12,24,27,32)(4,8,16,22,15,21,28,31)(7,18,20,26,19,29,30,13);(1,3,5,14,6,12,17,27)(2,7,9,20,10,19,23,30)(4,13,16,18,15,26,28,29)(8,11,22,25,21,24,31,32);(3,12)(4,15)(7,19)(11,24)(14,27)(16,28)(20,30)(25,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,f^2=h,b^a=bd,c^a=ce,g^a=gh,c^b=ch,e^b=egh,d^c=dg,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12963, #494361, #494362, … |
| maximal normal subgroup of | C2×#12963, #494361, #494362, … |
| maximal quotient of | #135954, #135955, #135966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5951);