#6579419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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	#61083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(9,10,15,22)(11,23,13,19)(12,18,31,38)(14,21,32,20)(16,27,35,26)(17,30,37,29)(24,40,28,34)(25,39,33,36);(2,5)(6,8)(9,11,14,26,15,13,32,27)(10,16,20,23,22,35,21,19)(12,25,29,28,31,33,30,24)(17,36,18,34,37,39,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,29,15,31,32,30)(10,17,20,18,22,37,21,38)(11,24,26,25,13,28,27,33)(16,34,23,39,35,40,19,36);(9,12)(10,18)(11,25)(13,33)(14,29)(15,31)(16,34)(17,21)(19,36)(20,37)(22,38)(23,39)(24,27)(26,28)(30,32)(35,40);(9,13)(10,19)(11,15)(12,28)(14,27)(16,20)(17,36)(18,34)(21,35)(22,23)(24,31)(25,30)(26,32)(29,33)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,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=eh,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, 6486615);