#109701

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

503 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)(4,12)(5,8)(6,10)(7,13)(11,15)(14,16)(17,18,22,25,23,19,21,27)(20,24,29,31,30,26,28,32);(1,3)(2,6,8,14)(4,10)(5,11)(7,9,12,15)(13,16)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,31,29,32);(4,13)(7,12)(9,15)(10,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,26,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16897);