#10666

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:

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

Polycyclic presentation:

2^8|a^2=dg,b^2=de,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	#94726, #94727, #126473, …
maximal normal subgroup of	#94726, #94727, #126473, …
maximal quotient of	#105230, #105235, #105241, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3654);