#6579431

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
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

1875 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)(9,10,16,20)(11,14,23,15)(12,18,26,33)(13,19,29,37)(17,30,21,25)(22,28,36,31)(24,34,40,38)(27,35,39,32);(2,5)(6,8)(9,11,14,20,16,23,15,10)(12,17,25,18,26,21,30,33)(13,19,28,22,29,37,31,36)(24,34,39,35,40,38,27,32);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,26,15,30)(10,17,11,18,20,21,23,33)(13,24,28,39,29,40,31,27)(19,32,22,34,37,35,36,38);(9,12)(10,18)(11,21)(13,27)(14,25)(15,30)(16,26)(17,23)(19,35)(20,33)(22,38)(24,28)(29,39)(31,40)(32,37)(34,36);(9,13)(10,19)(11,22)(12,24)(14,28)(15,31)(16,29)(17,32)(18,34)(20,37)(21,35)(23,36)(25,39)(26,40)(27,30)(33,38)

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,d^b=di,e^b=eh,h^b=hi,d^c=dhi,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, 6486627);