#6579437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
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	#61090	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)(11,22)(12,18)(13,20)(14,19)(15,21)(16,26)(17,31)(23,33)(24,39)(25,28)(27,35)(29,36)(30,38)(32,37)(34,40);(2,5)(6,8)(9,11,14,27,15,28,13,26)(10,16,20,25,21,35,19,22)(12,29,31,40,32,24,30,23)(17,36,18,33,38,39,37,34);(1,3,4,7)(2,6,5,8)(9,12,13,30,15,32,14,31)(10,17,19,37,21,38,20,18)(11,23,26,24,28,40,27,29)(16,33,22,36,35,34,25,39);(9,12)(10,18)(11,24)(13,30)(14,31)(15,32)(16,34)(17,19)(20,38)(21,37)(22,39)(23,27)(25,36)(26,40)(28,29)(33,35);(9,10)(11,25)(12,17)(13,19)(14,20)(15,21)(16,26)(18,31)(22,28)(23,39)(24,33)(27,35)(29,34)(30,37)(32,38)(36,40)

Polycyclic presentation:

2^9|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=ehi,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, 6486633);