#201136

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=hi,g^2=h,b^a=be,c^a=cf,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108332);