#483127

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	#60648	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

1015 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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, 390323);