#109459

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

588 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:

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

Polycyclic presentation:

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

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, 16655);