#6491449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470019
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398645);