#7142806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

1908 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=h,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=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, 7050002);