#6585062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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

1844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,7)(4,8)(5,6)(9,10,14,13)(11,20,12,16)(15,19,21,18)(17,22,24,23);(2,6)(7,8)(9,11,14,12)(10,16,13,20)(15,22,21,23)(17,18,24,19);(9,12,15,23,14,11,21,22)(10,17,18,16,13,24,19,20);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,14)(13,20)(15,23)(17,19)(18,24)(21,22);(1,4,5,3)(2,8,6,7)(9,13,14,10)(11,17,12,24)(15,19,21,18)(16,23,20,22)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72226931

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492258);