#7143211

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

1914 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)(11,22)(12,18)(13,21)(14,20)(15,19)(16,25)(17,27)(23,31)(24,38)(26,37)(28,36)(29,33)(30,34)(32,40)(35,39);(2,5)(3,7)(9,11)(10,16)(12,23)(13,29)(14,25)(15,26)(17,31)(18,35)(19,33)(20,37)(21,22)(24,27)(28,40)(30,39)(32,34)(36,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,23)(13,27)(14,30)(15,28)(16,31)(18,20)(19,34)(21,36)(22,38)(24,29)(25,39)(26,40)(32,33)(35,37);(2,5)(6,8)(9,12)(10,18)(11,24)(13,30)(14,27)(15,28)(16,32)(17,20)(19,36)(21,34)(22,38)(23,29)(25,40)(26,39)(31,33)(35,37);(1,3,4,7)(2,6,5,8)(10,19)(16,33)(17,34)(18,36)(20,21)(22,37)(31,32)(35,38)

Polycyclic presentation:

2^9|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, 7050407);