Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10,12,16)(11,21,22,50)(13,29,26,51)(14,34,27,59)(15,36,28,20)(17,41,38,23)(18,44,39,32)(19,47,40,31)(24,55,53,72)(25,56,37,52)(30,48,57,70)(33,66,58,62)(35,45,60,69)(42,67,68,64)(43,54,49,63)(46,65,61,71);(1,3,4,8)(2,5,6,7)(10,16)(17,38)(18,39)(19,40)(20,36)(21,50)(29,51)(34,59)(42,68)(43,49)(45,69)(46,61)(48,70)(52,56)(55,72)(62,66);(9,11,12,22)(10,17,16,38)(13,30,26,57)(14,35,27,60)(15,37,28,25)(18,45,39,69)(19,48,40,70)(20,49,36,43)(21,51,50,29)(23,44,41,32)(24,47,53,31)(33,67,58,64)(34,42,59,68)(46,72,61,55)(52,62,56,66)(54,65,63,71)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13708, #512960, #512962, … |
| maximal normal subgroup of | C2×#13708, #512960, #512962, … |
| maximal quotient of | #152272, #152434, #152436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6696);