Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C22×C4×#442 (order 1024) · Inn(G) = D8 · Out(G) = C22×C42 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C82 | C22 |
| socle | C22 | D4⋊C8 |
| prime radical | C2×D8 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)Polycyclic presentation:
2^8|a^2=d,c^2=eg,d^2=f,e^2=g,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=ce,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7068, #122440, #122441, … |
| maximal normal subgroup of | C2×#7068, #122440, #122441, … |
| maximal quotient of | #92821, #92869, #92931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56);