#6686578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18,23,22)(19,26,30,25)(20,31,28,29)(21,32,24,27);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19,23,30)(18,25,22,26)(20,21,28,24)(27,29,32,31);(17,20,21,19,23,28,24,30)(18,26,27,31,22,25,32,29);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,24)(18,27,22,32)(19,28,30,20)(25,29,26,31);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,23,18)(19,29,30,31)(20,25,28,26)(21,32,24,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

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, 6593774);