#483182

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60714	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390378);