#211721

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=i,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fi

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, 118917);