#6579423

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	#61088	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)(12,29)(13,19)(14,21)(15,20)(16,22)(17,26)(18,31)(24,34)(25,37)(27,36)(28,35)(30,33)(32,40)(38,39);(2,5)(6,8)(9,11,15,27,16,28,14,26)(10,17,21,35,22,36,20,23)(12,25,31,39,13,32,30,24)(18,37,29,34,33,40,19,38);(1,3,4,7)(2,6,5,8)(9,12,14,30,16,13,15,31)(10,18,20,19,22,33,21,29)(11,24,26,32,28,39,27,25)(17,34,23,37,36,38,35,40);(9,13,16,12)(10,19,22,29)(11,25,28,32)(14,31,15,30)(17,34,36,38)(18,20,33,21)(23,37,35,40)(24,27,39,26);(9,10)(11,23)(12,18)(13,33)(14,20)(15,21)(16,22)(17,27)(19,30)(24,37)(25,34)(26,36)(28,35)(29,31)(32,38)(39,40)

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,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, 6486619);