#7149014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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
prime radical	#60832	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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)(9,10,13,29,15,20,14,19)(11,21,24,16,26,37,25,32)(12,18,30,17,28,33,27,34)(22,31,39,36,38,40,23,35);(2,5)(3,7)(9,11,14,25,15,26,13,24)(10,16,29,37,20,32,19,21)(12,22,30,39,28,38,27,23)(17,31,18,35,34,40,33,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,31)(18,29)(19,33)(20,34)(21,36)(23,24)(25,39)(26,38)(32,40)(35,37);(2,5)(6,8)(9,12)(10,18)(11,23)(13,30)(14,27)(15,28)(16,31)(17,29)(19,34)(20,33)(21,35)(22,24)(25,38)(26,39)(32,40)(36,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^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=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, 7056210);