#198260

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1115 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105456);