#7063113

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	#61195	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(9,10,15,19,16,12,14,26)(11,20,24,17,23,33,30,21)(13,18,28,36,29,25,31,35)(22,37,40,32,39,34,27,38);(1,2)(3,7)(4,6)(5,8)(9,11,16,23)(10,17,12,21)(13,27,29,40)(14,30,15,24)(18,34,25,37)(19,33,26,20)(22,31,39,28)(32,36,38,35);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,21)(13,25)(14,26)(15,19)(17,23)(18,29)(20,30)(22,38)(24,33)(27,37)(28,36)(31,35)(32,39)(34,40);(3,8)(5,7)(9,13,16,29)(10,18,12,25)(11,22,23,39)(14,31,15,28)(17,32,21,38)(19,36,26,35)(20,37,33,34)(24,40,30,27);(2,6)(5,7)(11,23)(17,21)(20,33)(22,39)(24,30)(27,40)(32,38)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6970309);