#483165

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	#60717	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

733 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)(9,10,14,20,32,23,40,37,15,21,34,19,13,18,31,38)(11,22,25,29,17,35,33,30,26,39,24,12,27,28,36,16);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,23)(13,17)(14,33)(15,26)(18,29)(19,28)(20,35)(21,30)(22,37)(24,31)(25,40)(27,32)(34,36)(38,39);(1,3,5,4)(2,6,7,8)(9,12,15,29)(10,17,21,27)(11,23,26,18)(13,30,32,16)(14,28,34,35)(19,36,20,33)(22,31,39,40)(24,38,25,37);(9,13,15,32)(10,18,21,23)(11,17,26,27)(12,16,29,30)(14,31,34,40)(19,37,20,38)(22,35,39,28)(24,36,25,33)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=hi,f^2=gh,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, 390361);