#6584347

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	#60460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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)(9,10,13,19)(11,17,23,15)(12,18,28,36)(14,21,31,20)(16,25,34,24)(22,35,39,32)(26,40,27,33)(29,38,30,37);(1,3,5,4)(2,6,7,8)(9,11,13,23)(10,15,19,17)(12,26,28,27)(14,24,31,25)(16,21,34,20)(18,33,36,40)(22,30,39,29)(32,37,35,38);(9,11,14,24,13,23,31,25)(10,16,20,17,19,34,21,15)(12,22,29,27,28,39,30,26)(18,33,37,35,36,40,38,32);(1,4,5,3)(2,6,7,8)(9,11,13,23)(10,17,19,15)(12,27,28,26)(14,24,31,25)(16,20,34,21)(18,33,36,40)(22,29,39,30)(32,37,35,38);(9,12)(10,18)(11,22)(13,28)(14,29)(15,32)(16,33)(17,35)(19,36)(20,37)(21,38)(23,39)(24,27)(25,26)(30,31)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6491543);