#114909

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22105);