#211709

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
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

629 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:

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

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^c=di,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118905);