#168416

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37219	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75612);