#7143629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2144 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)(11,20)(12,17)(13,29)(14,18)(15,24)(16,27)(19,30)(21,31)(22,36)(23,37)(25,32)(26,34)(28,35)(33,38)(39,40);(2,5)(3,7)(9,11,13,24,14,23,30,38)(10,15,19,20,18,33,29,37)(12,25,26,22,28,21,27,39)(16,32,35,40,34,31,17,36);(1,2)(3,6)(4,5)(7,8)(9,12,14,28)(10,16,18,34)(11,21,23,25)(13,26,30,27)(15,31,33,32)(17,19,35,29)(20,36,37,40)(22,24,39,38);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,32)(16,29)(18,35)(19,34)(20,36)(21,38)(23,39)(24,25)(26,30)(31,33)(37,40);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(16,34)(17,35)(19,29)(21,25)(22,39)(24,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050825);