Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×C8×D4 (order 512) · Inn(G) = D4 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Q8⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C2×C32 |
| Frattini, cosocle | C4×C16 | C22 |
| socle, prime radical | C22 | C22⋊C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40);(1,2,4,6)(3,8,7,5)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=e,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,c^b=ce112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×Q8⋊C32, #150076, #150077, … |
| maximal normal subgroup of | C2×Q8⋊C32, #150076, #150077, … |
| maximal quotient of | #93927, #93967, #94036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 380);