#7063143

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

1818 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)(3,5)(4,6)(7,8)(9,10,14,19,15,20,13,28)(11,21,25,16,24,34,29,33)(12,18,26,36,27,35,30,17)(22,32,39,40,38,31,23,37);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,20,33)(12,22,27,38)(13,29,14,25)(17,31,36,32)(18,37,35,40)(19,34,28,21)(23,26,39,30);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,26)(14,30)(15,27)(16,31)(18,28)(19,35)(20,36)(21,37)(23,25)(24,38)(29,39)(32,33)(34,40);(3,8)(5,7)(9,12)(10,18)(11,23)(13,30)(14,26)(15,27)(16,32)(17,28)(19,36)(20,35)(21,37)(22,25)(24,39)(29,38)(31,33)(34,40);(2,6)(5,7)(11,24)(16,33)(21,34)(22,38)(23,39)(25,29)(31,32)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,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, 6970339);