#6575791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

2058 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)(11,23)(12,27)(13,18)(14,20)(15,21)(16,33)(17,22)(19,31)(24,36)(25,38)(26,34)(28,40)(29,35)(30,39)(32,37);(2,5)(6,8)(9,11,15,13)(10,18,21,23)(12,28,30,14)(16,26,17,29)(19,36,38,37)(20,39,40,27)(22,34,33,35)(24,31,32,25);(9,12,15,30)(10,19,21,38)(11,24,13,32)(14,29,28,26)(16,25,17,31)(18,20,23,40)(22,39,33,27)(34,36,35,37);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,18,21,23)(12,28,30,14)(16,29,17,26)(19,37,38,36)(20,27,40,39)(22,34,33,35)(24,31,32,25);(9,14,15,28)(10,20,21,40)(11,25,13,31)(12,29,30,26)(16,24,17,32)(18,19,23,38)(22,37,33,36)(27,35,39,34)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,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, 6482987);