#6599385

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,25)(19,26,29,32)(20,24,22,31)(23,28,27,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,31)(21,25)(22,24)(23,29)(26,28)(30,32);(11,12)(14,16)(17,19,21,29)(18,23,25,27)(20,30,22,28)(24,32,31,26);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,24,25,31)(19,28,29,30)(23,26,27,32);(11,12)(14,16)(17,20,21,22)(18,24,25,31)(19,28,29,30)(23,26,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ehi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506581);