#483185

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 390381);