#6470831

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60755	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378027);