#10667

order	256
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#399
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,7,8,5)(9,10,13,18,15,19,14,28)(11,20,22,31,23,36,24,16)(12,17,26,33,27,35,29,34)(21,32,25,30,38,37,39,40);(1,2)(3,5)(4,7)(6,8)(9,11,14,22,15,23,13,24)(10,16,28,20,19,31,18,36)(12,25,29,38,27,39,26,21)(17,32,34,30,35,37,33,40);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,33)(19,35)(20,37)(22,25)(23,38)(24,39)(28,34)(31,40)(32,36)

Polycyclic presentation:

2^8|a^2=d,b^2=deg,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=eh,f^c=fg

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94720, #94721, #126286, …
maximal normal subgroup of	#94720, #94721, #126286, …
maximal quotient of	#105229, #105230, #105231, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3655);