#6585826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493022);