#7142826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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)(9,10,15,19)(11,22,25,38)(12,27,30,18)(13,21,14,20)(16,31,35,26)(17,32,36,29)(23,33,28,34)(24,37,40,39);(2,5)(3,7)(9,11)(10,16)(12,28)(13,31)(14,26)(15,25)(17,34)(18,37)(19,35)(20,38)(21,22)(23,30)(24,29)(27,39)(32,40)(33,36);(1,2)(3,6)(4,5)(7,8)(9,12,15,30)(10,17,19,36)(11,23,25,28)(13,29,14,32)(16,33,35,34)(18,20,27,21)(22,39,38,37)(24,31,40,26);(2,5)(6,8)(9,12)(10,18)(11,24)(13,32)(14,29)(15,30)(16,34)(17,21)(19,27)(20,36)(22,39)(23,26)(25,40)(28,31)(33,35)(37,38);(1,3,4,7)(2,6,5,8)(10,19)(11,25)(17,36)(18,27)(20,21)(23,28)(24,40)(26,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050022);