#6492413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1879 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)(12,16,23,24)(19,21,20,22);(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)(9,12,14,22,15,23,13,21)(10,16,17,20,18,24,11,19);(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)(10,18)(11,17)(16,24)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6399609);