#6575444

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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

2078 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,26)(13,20)(14,21)(15,33)(16,22)(17,23)(19,28)(24,39)(25,36)(27,34)(29,38)(30,40)(31,37)(32,35);(2,5)(6,8)(9,11,17,25)(10,18,23,36)(12,27,30,31)(13,16,32,15)(14,29,24,28)(19,21,38,39)(20,22,35,33)(26,34,40,37);(9,12,17,30)(10,19,23,38)(11,24,25,14)(13,31,32,27)(15,29,16,28)(18,34,36,37)(20,39,35,21)(22,40,33,26);(1,3,4,7)(2,6,5,8)(9,13)(10,20)(11,16)(12,14)(15,25)(17,32)(18,22)(19,37)(21,26)(23,35)(24,30)(27,29)(28,31)(33,36)(34,38)(39,40);(9,14)(10,21)(11,12)(13,28)(15,31)(16,27)(17,24)(18,26)(19,20)(22,34)(23,39)(25,30)(29,32)(33,37)(35,38)(36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,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=dh,g^c=gi,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, 6482640);