#6584323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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	#60462	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1953 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)(11,22,24,17)(12,18,26,33)(13,19,28,36)(15,30,16,21)(23,35,39,32)(25,34,27,40)(29,38,31,37);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,20,22)(12,16,26,15)(13,27,28,25)(18,30,33,21)(19,34,36,40)(23,31,39,29)(32,37,35,38);(9,11,16,26,14,24,15,12)(10,18,30,17,20,33,21,22)(13,23,31,27,28,39,29,25)(19,34,38,32,36,40,37,35);(1,4,5,3)(2,6,7,8)(9,12,14,26)(10,18,20,33)(11,15,24,16)(13,23,28,39)(17,21,22,30)(19,35,36,32)(25,31,27,29)(34,37,40,38);(9,13)(10,19)(11,23)(12,25)(14,28)(15,29)(16,31)(17,32)(18,34)(20,36)(21,37)(22,35)(24,39)(26,27)(30,38)(33,40)

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,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491519);