#6586315

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493511);