#6676836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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

1805 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)(3,6)(4,8)(5,7)(9,10)(11,21)(12,27)(13,20)(14,15)(16,19)(17,25)(18,29)(22,38)(23,24)(26,34)(28,33)(30,32)(31,36)(35,40)(37,39);(1,3,6,2)(4,8,5,7)(9,11,15,25)(10,17,14,21)(12,28,31,39)(13,23,16,26)(18,35,30,38)(19,24,20,34)(22,32,40,29)(27,37,36,33);(9,12,15,31)(10,18,14,30)(11,22,25,40)(13,29,16,32)(17,33,21,37)(19,27,20,36)(23,39,26,28)(24,35,34,38);(1,4,6,5)(2,7,3,8)(9,13,15,16)(10,19,14,20)(11,23,25,26)(12,29,31,32)(17,24,21,34)(18,27,30,36)(22,39,40,28)(33,35,37,38);(1,5,6,4)(2,7,3,8)(9,14,15,10)(11,24,25,34)(12,30,31,18)(13,20,16,19)(17,23,21,26)(22,35,40,38)(27,29,36,32)(28,33,39,37)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584032);