#7003082

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

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,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,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, 6910278);