#6584349

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

1968 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)(11,15,25,33)(12,17,30,37)(13,19,21,18)(16,24,32,23)(22,31,39,40)(26,35,27,34)(28,38,29,36);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,26)(13,23)(14,25)(16,19)(17,35)(18,32)(20,33)(21,24)(22,28)(27,30)(29,39)(31,38)(34,37)(36,40);(9,11,21,24,14,25,13,23)(10,16,19,33,20,32,18,15)(12,22,29,26,30,39,28,27)(17,34,38,40,37,35,36,31);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,27)(13,23)(14,25)(16,19)(17,34)(18,32)(20,33)(21,24)(22,29)(26,30)(28,39)(31,36)(35,37)(38,40);(9,12)(10,17)(11,22)(13,28)(14,30)(15,31)(16,34)(18,36)(19,38)(20,37)(21,29)(23,27)(24,26)(25,39)(32,35)(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,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, 6491545);