#6579436

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61088	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(11,19)(12,18)(13,21)(14,20)(15,22)(16,25)(17,29)(23,31)(24,34)(26,33)(27,32)(28,36)(30,35)(37,39)(38,40);(2,5)(6,8)(9,11,13,25,15,27,14,26)(10,16,20,19,22,33,21,32)(12,24,28,23,30,37,29,38)(17,34,35,40,36,39,18,31);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,30,14,29)(10,17,20,35,22,36,21,18)(11,23,25,37,27,38,26,24)(16,31,19,34,33,40,32,39);(9,12,15,30)(10,18,22,35)(11,24,27,37)(13,28,14,29)(16,31,33,40)(17,21,36,20)(19,34,32,39)(23,26,38,25);(9,11)(10,19)(12,23)(13,25)(14,26)(15,27)(16,21)(17,34)(18,31)(20,33)(22,32)(24,29)(28,37)(30,38)(35,40)(36,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,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, 6486632);