#6355521

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	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1767 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)(13,20)(14,19)(15,18)(16,24)(17,27)(22,36)(23,32)(25,31)(28,34)(29,33)(30,37)(35,38)(39,40);(1,3,5,2,6,7,4,8)(9,11,13,24,15,23,14,25)(10,16,19,21,18,31,20,32)(12,22,27,38,28,37,29,39)(17,30,26,35,33,36,34,40);(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);(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39);(1,4,6,5)(2,7,8,3)(10,18)(11,23)(17,33)(19,20)(22,37)(24,25)(26,34)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,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, 6262717);