#6583578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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

2079 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)(9,10,14,20)(11,18,12,16)(13,19,30,38)(15,22,26,21)(17,24,34,25)(23,36,27,33)(28,35,29,37)(31,40,32,39);(2,5)(6,8)(9,11)(10,16)(12,14)(13,28)(15,24)(17,22)(18,20)(19,37)(21,34)(23,31)(25,26)(27,32)(29,30)(33,40)(35,38)(36,39);(9,12,26,24,14,11,15,25)(10,17,22,18,20,34,21,16)(13,27,32,29,30,23,31,28)(19,35,40,36,38,37,39,33);(1,3,4,7)(2,6,5,8)(9,11,14,12)(10,18,20,16)(13,29,30,28)(15,24,26,25)(17,22,34,21)(19,37,38,35)(23,31,27,32)(33,39,36,40);(9,13)(10,19)(11,23)(12,27)(14,30)(15,31)(16,33)(17,35)(18,36)(20,38)(21,39)(22,40)(24,29)(25,28)(26,32)(34,37)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,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=ehi,h^b=hi,d^c=dh,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, 6490774);