#6579434

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	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(12,18,32,38)(14,21,24,20)(16,28,35,27)(17,31,37,30)(25,40,29,34)(26,39,33,36);(2,5)(6,8)(9,11,24,28,15,13,14,27)(10,16,21,19,22,35,20,23)(12,26,31,25,32,33,30,29)(17,36,38,40,37,39,18,34);(1,3,4,7)(2,6,5,8)(9,12,14,30,15,32,24,31)(10,17,20,18,22,37,21,38)(11,25,27,26,13,29,28,33)(16,34,23,39,35,40,19,36);(9,12)(10,18)(11,26)(13,33)(14,30)(15,32)(16,34)(17,21)(19,36)(20,37)(22,38)(23,39)(24,31)(25,28)(27,29)(35,40);(9,13)(10,19)(11,15)(12,29)(14,28)(16,20)(17,36)(18,34)(21,35)(22,23)(24,27)(25,32)(26,31)(30,33)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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=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, 6486630);