#6542235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#62957	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3004 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)(11,20,24,31)(12,25,28,33)(13,29)(14,18)(15,19)(16,22,32,23)(17,26,34,27)(21,36)(30,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,24)(10,16,19,32)(12,21,28,38)(13,22,14,23)(17,30,34,36)(18,31,29,20)(25,35,33,40)(26,37,27,39);(2,5)(6,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,38)(25,29)(31,40)(32,36);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,31,32,20)(17,33,34,25)(21,37,38,39)(30,40,36,35);(3,7)(6,8)(10,19)(16,32)(17,34)(18,29)(20,31)(25,33)(30,36)(35,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,f^b=fi,g^c=gi,e^d=eh

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, 6449431);