Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | Aut(G) = C2×#72225051 (order 2048) · Inn(G) = #357 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #357 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C22×C16 | C22 |
| prime radical | C23 | D16 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(10,16,17,32)(11,21)(12,27)(13,23)(14,29)(18,35,19,36)(20,37,31,34)(22,28,33,30)(24,26)(25,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,15,17)(11,22,26,33)(12,28,13,30)(14,31,29,20)(16,21,32,24)(18,23,19,27)(25,35,38,36)(34,40,37,39)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fg,e^2=gh,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7419, #150242, #150243, … |
| maximal normal subgroup of | C2×#7419, #150242, #150243, … |
| maximal quotient of | #93785, #93806, #93851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 407);