#6491807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470409
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1997 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:

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

Polycyclic presentation:

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

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, 6399003);