Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×C8×AGL(1,ℤ/8) (order 512) · Inn(G) = D8 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D8 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | D4⋊C16 |
| prime radical | #135 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,30,42,6,14,29,43,4,23,26,54,8,15,31,44,27,10,37,39,25,12,45,41,28,40,56,62)(3,16,18,47,20,36,51,60,21,38,52,61,24,46,55,63,22,48,53,64,17,9,32,33,19,35,50,59,49,34,57,58);(1,3)(2,9)(4,24)(5,18)(6,21)(7,20)(8,22)(10,38)(11,33)(12,46)(13,35)(14,34)(15,36)(16,23)(17,27)(19,25)(26,55)(28,49)(29,52)(30,51)(31,53)(32,37)(39,61)(40,48)(41,63)(42,59)(43,58)(44,60)(45,50)(47,54)(56,57)(62,64)Polycyclic presentation:
2^8|a^2=d,c^2=eh,d^2=f,e^2=h,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=ce,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7381, #150051, #150107, … |
| maximal normal subgroup of | C2×#7381, #150051, #150107, … |
| maximal quotient of | #93745, #93753, #93756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 369);