#110693

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

656 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)(3,9,12,7)(4,13)(5,14)(6,8)(10,11,16,15)(17,18,20,24,22,26,21,25)(19,27,29,32,30,31,28,23);(1,3)(2,7)(4,10)(5,15)(6,12)(8,9)(11,13)(14,16)(17,19,21,28,22,30,20,29)(18,23,25,31,26,32,24,27);(1,4,13,2,6,14,5,8)(3,10,11,7,12,16,15,9)(18,24,26,25)(23,27,32,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17889);