#6354771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1785 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:

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

Polycyclic presentation:

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

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, 6261967);