#6579428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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

1894 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)(11,18)(12,17)(13,20)(14,21)(15,24)(16,27)(19,30)(22,36)(23,34)(25,32)(26,38)(28,35)(29,37)(31,39)(33,40);(2,5)(6,8)(9,11,13,24,14,26,30,25)(10,15,19,18,21,32,20,38)(12,23,27,39,29,40,28,22)(16,33,17,31,35,34,37,36);(1,3,4,7)(2,6,5,8)(9,12,13,27,14,29,30,28)(10,16,19,17,21,35,20,37)(11,22,24,23,26,39,25,40)(15,31,18,34,32,36,38,33);(9,12,14,29)(10,17,21,37)(11,23,26,40)(13,27,30,28)(15,31,32,36)(16,19,35,20)(18,34,38,33)(22,24,39,25);(9,11)(10,18)(12,22)(13,24)(14,26)(15,20)(16,34)(17,36)(19,32)(21,38)(23,27)(25,30)(28,40)(29,39)(31,37)(33,35)

Polycyclic presentation:

2^9|b^2=g,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=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486624);