#6584322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^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=eh,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

#72226912, #72226925, #72226937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491518);