#483184

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	#60718	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

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,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, 390380);