#211933

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#19128	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119129);