#6435412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2950 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:

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

Polycyclic presentation:

2^9|a^2=fg,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi

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, 6342608);