#201109

order	512
nilpotency class	3
isoclinism	isoclinic class of #201103
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108305);