#7143222

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

1893 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)(9,10,16,22)(11,23,26,38)(12,28,13,19)(14,21,15,20)(17,32,35,27)(18,31,36,30)(24,33,29,34)(25,39,40,37);(2,5)(3,7)(9,11)(10,17)(12,29)(13,24)(14,32)(15,27)(16,26)(18,34)(19,37)(20,38)(21,23)(22,35)(25,31)(28,39)(30,40)(33,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,24)(13,16)(14,30)(15,31)(17,33)(19,21)(20,28)(22,36)(23,39)(25,32)(26,29)(27,40)(34,35)(37,38);(2,5)(6,8)(9,13,16,12)(10,19,22,28)(11,25,26,40)(14,30,15,31)(17,34,35,33)(18,20,36,21)(23,39,38,37)(24,27,29,32);(1,3,4,7)(2,6,5,8)(11,26)(17,35)(23,38)(24,29)(25,40)(27,32)(33,34)(37,39)

Polycyclic presentation:

2^9|a^2=i,d^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, 7050418);