#7149420

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
prime radical	#60834	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 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)(9,10,14,19,15,20,13,30)(11,21,26,33,24,39,25,16)(12,18,28,36,29,35,31,17)(22,32,23,38,27,34,40,37);(2,5)(3,7)(9,11,14,26,15,24,13,25)(10,16,30,39,20,33,19,21)(12,27,31,23,29,22,28,40)(17,34,18,37,36,32,35,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,29)(10,17,20,36)(11,22,24,27)(13,28,14,31)(16,32,33,34)(18,19,35,30)(21,37,39,38)(23,26,40,25);(2,5)(6,8)(9,12)(10,18)(11,23)(13,31)(14,28)(15,29)(16,32)(17,30)(19,36)(20,35)(21,38)(22,25)(24,40)(26,27)(33,34)(37,39);(1,3,4,7)(2,6,5,8)(11,24)(16,33)(21,39)(22,27)(23,40)(25,26)(32,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,e^2=hi,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, 7056616);