#7148517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055713);