#146331

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₈
commutator	C₈	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 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)(11,22)(12,18)(13,19)(14,20)(15,30)(16,21)(17,23)(24,36)(25,31)(26,40)(27,37)(28,34)(29,35)(32,38)(33,39);(2,4)(9,11,13,24,15,26,29,39,16,27,28,38,14,25,12,23)(10,17,18,31,20,32,34,37,21,33,35,40,30,36,19,22);(9,12,14,28,16,29,15,13)(10,18,20,34,21,35,30,19)(11,23,25,38,27,39,26,24)(17,31,32,37,33,40,36,22)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi

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, 53527);