#7148516

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	#60933	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(9,10,14,19,15,20,13,29)(11,21,25,16,24,35,30,34)(12,18,27,37,28,36,31,17)(22,33,40,38,26,32,23,39);(2,5)(3,7)(9,11,15,24)(10,16,20,34)(12,26,28,22)(13,30,14,25)(17,33,37,32)(18,38,36,39)(19,35,29,21)(23,31,40,27);(1,2)(3,6)(4,5)(7,8)(9,12,15,28)(10,17,20,37)(11,22,24,26)(13,27,14,31)(16,32,34,33)(18,19,36,29)(21,39,35,38)(23,30,40,25);(2,5)(6,8)(9,12)(10,18)(11,23)(13,31)(14,27)(15,28)(16,33)(17,29)(19,37)(20,36)(21,39)(22,25)(24,40)(26,30)(32,34)(35,38);(1,3,4,7)(2,6,5,8)(11,24)(16,34)(21,35)(22,26)(23,40)(25,30)(32,33)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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,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, 7055712);