#7143636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2124 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)(9,10,16,20)(11,14,19,15)(12,21,26,23)(13,18,30,36)(17,31,33,25)(22,28,35,29)(24,37,27,38)(32,40,34,39);(2,5)(3,7)(9,11,14,20,16,19,15,10)(12,23,25,17,26,21,31,33)(13,18,29,35,30,36,28,22)(24,34,40,37,27,32,39,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,26)(10,17,20,33)(11,21,19,23)(13,27,30,24)(14,25,15,31)(18,34,36,32)(22,38,35,37)(28,40,29,39);(2,5)(6,8)(9,13)(10,18)(11,22)(12,24)(14,28)(15,29)(16,30)(17,32)(19,35)(20,36)(21,37)(23,38)(25,39)(26,27)(31,40)(33,34);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,20,11)(12,25,26,31)(13,28,30,29)(17,23,33,21)(18,35,36,22)(24,39,27,40)(32,38,34,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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,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, 7050832);