#6357541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh

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, 6264737);