#7148113

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

1690 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)(9,10,15,19,16,20,14,31)(11,21,23,17,12,24,26,34)(13,27,29,36,30,18,32,37)(22,35,39,40,25,38,28,33);(2,5)(3,7)(9,11,16,12)(10,17,20,34)(13,28,30,39)(14,26,15,23)(18,35,27,38)(19,24,31,21)(22,32,25,29)(33,37,40,36);(1,2)(3,6)(4,5)(7,8)(9,12,16,11)(10,17,20,34)(13,25,30,22)(14,26,15,23)(18,33,27,40)(19,21,31,24)(28,29,39,32)(35,37,38,36);(2,5)(6,8)(9,13)(10,18)(11,22)(12,25)(14,32)(15,29)(16,30)(17,33)(19,37)(20,27)(21,38)(23,39)(24,35)(26,28)(31,36)(34,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,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, 7055309);