#6355550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
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
prime radical	#60546	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1759 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)(11,21)(12,26)(13,18)(14,20)(15,19)(16,25)(17,27)(22,36)(23,32)(24,31)(28,33)(29,34)(30,38)(35,39)(37,40);(1,3,5,2,6,7,4,8)(9,11,13,25,15,24,14,23)(10,16,20,21,19,32,18,31)(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)(19,34)(20,26)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(9,13,15,14)(10,18,19,20)(11,23,24,25)(12,27,28,29)(16,21,32,31)(17,33,34,26)(22,37,38,39)(30,35,36,40);(1,4,6,5)(2,7,8,3)(10,19)(11,24)(17,34)(18,20)(22,38)(23,25)(26,33)(37,39)

Polycyclic presentation:

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

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, 6262746);