#6587686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,24,28,29)(21,25,31,27)(22,26,32,30);(2,6)(7,8)(17,19,23,28)(18,24,20,29)(21,26,31,30)(22,25,32,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,18)(19,21,28,31)(22,26,32,30)(24,25,29,27);(9,11,12,15)(10,14,13,16)(17,21,23,31)(18,25,20,27)(19,26,28,30)(22,24,32,29);(1,4,5,3)(2,8,6,7)(19,25)(21,29)(22,32)(24,31)(26,30)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494882);