#7143224

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

1922 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)(9,10,15,21)(11,22,25,38)(12,18,29,37)(13,20,14,19)(16,30,33,26)(17,31,35,28)(23,32,27,34)(24,39,40,36);(2,5)(3,7)(9,11)(10,16)(12,27)(13,30)(14,26)(15,25)(17,34)(18,36)(19,38)(20,22)(21,33)(23,29)(24,28)(31,40)(32,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,29)(10,17,21,35)(11,23,25,27)(13,28,14,31)(16,32,33,34)(18,20,37,19)(22,36,38,39)(24,30,40,26);(2,5)(6,8)(9,12)(10,18)(11,24)(13,31)(14,28)(15,29)(16,32)(17,19)(20,35)(21,37)(22,39)(23,26)(25,40)(27,30)(33,34)(36,38);(1,3,4,7)(2,6,5,8)(11,25)(16,33)(22,38)(23,27)(24,40)(26,30)(32,34)(36,39)

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,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, 7050420);