#148106

order	512
nilpotency class	4
isoclinism	isoclinic class of #135932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55302);