#6583571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
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

2093 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)(4,5)(7,8)(9,10)(11,22)(12,29)(13,18)(14,19)(15,21)(16,26)(17,30)(20,33)(23,37)(24,38)(25,28)(27,36)(31,34)(32,35)(39,40);(2,5)(6,8)(9,11,14,26)(10,16,19,22)(12,24,13,32)(15,27,33,28)(17,37,34,40)(18,38,29,35)(20,36,21,25)(23,30,39,31);(9,12,15,30,14,13,33,31)(10,17,20,29,19,34,21,18)(11,23,27,24,26,39,28,32)(16,35,36,40,22,38,25,37);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,18,19,29)(11,24,26,32)(15,31,33,30)(16,35,22,38)(17,21,34,20)(23,27,39,28)(25,37,36,40);(9,10)(11,25)(12,17)(13,34)(14,19)(15,20)(16,27)(18,31)(21,33)(22,28)(23,37)(24,35)(26,36)(29,30)(32,38)(39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490767);