#6583567

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

2091 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)(12,16)(13,18)(14,19)(15,21)(17,24)(20,26)(23,40)(25,35)(27,33)(28,34)(29,36)(30,37)(31,39)(32,38);(2,5)(6,8)(9,11,14,12)(10,16,19,22)(13,28,30,29)(15,24,26,25)(17,21,35,20)(18,36,37,34)(23,31,27,32)(33,39,40,38);(9,11,15,24,14,12,26,25)(10,17,20,22,19,35,21,16)(13,23,31,28,30,27,32,29)(18,34,38,40,37,36,39,33);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,16,19,22)(13,29,30,28)(15,25,26,24)(17,21,35,20)(18,36,37,34)(23,32,27,31)(33,39,40,38);(9,13)(10,18)(11,23)(12,27)(14,30)(15,31)(16,33)(17,34)(19,37)(20,38)(21,39)(22,40)(24,28)(25,29)(26,32)(35,36)

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,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, 6490763);