#6470819

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

1932 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)(9,10,15,19)(11,20,18,14)(12,23,26,16)(13,17,27,31)(21,25,30,33)(22,34,32,28)(24,36,37,29)(35,38,39,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,10,15,18,20,19)(12,21,25,16,26,30,33,23)(13,22,28,17,27,32,34,31)(24,35,38,29,37,39,40,36);(3,7)(5,8)(9,12,14,25,15,26,20,33)(10,16,18,30,19,23,11,21)(13,24,28,38,27,37,34,40)(17,29,32,39,31,36,22,35);(3,7)(5,8)(9,13)(10,17)(11,22)(12,24)(14,28)(15,27)(16,29)(18,32)(19,31)(20,34)(21,35)(23,36)(25,38)(26,37)(30,39)(33,40);(2,6)(5,8)(13,27)(17,31)(22,32)(24,37)(28,34)(29,36)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,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, 6378015);