#7143212

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

1915 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)(10,16)(11,20)(13,14)(18,32)(19,28)(22,37)(23,30)(24,36)(25,39)(26,29)(31,34)(38,40);(2,5)(3,7)(9,10)(11,21)(12,24)(13,28)(14,16)(15,19)(17,23)(18,26)(20,31)(22,38)(25,40)(27,32)(29,36)(30,34)(33,37)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,25)(13,20)(14,30)(15,23)(16,31)(18,35)(19,21)(22,27)(24,38)(26,37)(28,34)(29,39)(32,40)(33,36);(2,5)(6,8)(9,12)(10,18)(11,22)(13,29)(14,26)(15,27)(16,32)(17,33)(19,36)(20,37)(21,35)(23,25)(24,28)(30,39)(31,38)(34,40);(1,3,4,7)(2,6,5,8)(10,19)(16,28)(17,21)(18,36)(24,32)(31,34)(33,35)(38,40)

Polycyclic presentation:

2^9|e^2=hi,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, 7050408);