#7182013

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	#60926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=h,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, 7089209);