#114971

order	512
nilpotency class	3
isoclinism	isoclinic class of #9946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22167);