#7143219

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

1943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,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, 7050415);