#6513594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
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	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1915 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6420790);