#6579424

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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	#61087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(11,23)(12,30)(13,20)(14,17)(15,22)(16,21)(18,27)(19,33)(24,36)(25,37)(26,29)(28,35)(31,32)(34,39)(38,40);(2,5)(6,8)(9,11,15,27,17,29,16,28)(10,18,21,23,14,35,22,26)(12,25,32,24,13,34,33,40)(19,37,20,38,31,39,30,36);(1,3,4,7)(2,6,5,8)(9,12,15,32,17,13,16,33)(10,19,21,20,14,31,22,30)(11,24,27,34,29,40,28,25)(18,36,23,37,35,38,26,39);(9,13,17,12)(10,20,14,30)(11,25,29,34)(15,33,16,32)(18,36,35,38)(19,21,31,22)(23,37,26,39)(24,28,40,27);(9,14)(10,17)(11,26)(12,31)(13,19)(15,22)(16,21)(18,27)(20,33)(23,29)(24,39)(25,38)(28,35)(30,32)(34,36)(37,40)

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,d^b=di,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, 6486620);