#6580518

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,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, 6487714);