#483126

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	#60647	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

1083 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:

(1,2,3,6,4,5,7,8)(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);(2,5)(3,7)(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);(2,5)(6,8)(9,11)(10,17)(12,23)(13,24)(14,22)(15,25)(16,30)(18,33)(19,21)(20,34)(26,37)(27,32)(28,40)(29,39)(31,36)(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,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, 390322);