#6686123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593319);