#110847

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18043);