#147146

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 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)(3,4)(5,6)(7,8)(9,10,13,19,14,18,26,33,15,20,27,34,12,16,25,29)(11,17,22,31,23,30,36,39,24,32,37,40,21,28,35,38);(2,4)(10,16,20,18)(17,28,32,30)(19,29,34,33)(31,38,40,39);(6,8)(9,11,13,22,14,23,26,36,15,24,27,37,12,21,25,35)(10,17,19,31,18,30,33,39,20,32,34,40,16,28,29,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54342);