Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,22)(13,25)(14,29)(15,19)(17,23)(18,27)(20,36)(21,35)(24,39)(26,31)(28,38)(30,34)(32,40)(33,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,33,32)(18,25,35,38)(22,36,39,34)(23,26,37,40);(9,12,15,24)(10,17,19,33)(11,20,14,30)(13,26,28,40)(16,31,29,32)(18,34,35,36)(21,37,27,23)(22,38,39,25);(3,8)(5,7)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=de,f^c=fg,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33867, #7623683, #7623717, … |
| maximal normal subgroup of | C2×#33867, #7623683, #7623717, … |
| maximal quotient of | #497380, #497539, #497718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26855);