#6584329

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491525);