#6590150

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1936 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(18,23)(19,27)(24,29)(25,26);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,28,27,32)(20,29,30,24)(22,25,31,26);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(17,19,21,27)(18,20,23,30)(22,29,31,24)(25,28,26,32);(9,12,13,11)(10,16,15,14)(17,20)(18,24)(19,26)(21,30)(22,32)(23,29)(25,27)(28,31);(1,4,5,3)(2,8,6,7)(18,25)(20,28)(22,31)(23,26)(24,29)(30,32)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6497346);