Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 16384 |
| 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 | #2246 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 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:
(2,6)(7,8)(9,10,15,19)(11,16,14,29)(12,22,24,39)(13,25,28,38)(17,32,34,23)(18,21,36,27)(20,37,30,35)(26,31,40,33);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,34,33)(18,25,36,38)(22,37,39,35)(23,26,32,40);(1,3,5,4)(2,7,6,8)(9,12,15,24)(10,17,19,34)(11,20,14,30)(13,26,28,40)(16,31,29,33)(18,35,36,37)(21,32,27,23)(22,38,39,25);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=eh,d^2=e,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 | #151531, #151535, #151583, … |
| maximal normal subgroup of | #151531, #151535, #151583, … |
| maximal quotient of | #497667, #497935, #498049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26951);