#148711

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^c=ehi,g^c=gi,h^c=hi

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, 55907);