#109703

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,14,18,15,19,13,27)(11,20,24,30,23,35,22,16)(12,17,25,32,26,33,28,31)(21,36,37,29,38,34,39,40);(2,5)(3,8)(6,7)(9,11)(10,16,19,30)(12,21)(13,22)(14,24)(15,23)(17,29,33,40)(18,20,27,35)(25,37)(26,38)(28,39)(31,36,32,34);(9,12,14,25,15,26,13,28)(10,17,18,32,19,33,27,31)(11,21,24,37,23,38,22,39)(16,29,20,34,30,40,35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,f^2=hi,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, 16899);