#6440841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2580 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)(9,10,15,19)(11,20,23,27)(12,26,18,14)(13,17,32,36)(16,25,21,24)(22,37,39,34)(28,31,35,30)(29,33,40,38);(2,5)(6,8)(9,11,14,24,15,23,26,25)(10,16,18,20,19,21,12,27)(13,29,30,39,32,40,31,22)(17,34,35,33,36,37,28,38);(1,3,4,7)(2,6,5,8)(9,12,14,10,15,18,26,19)(11,21,24,27,23,16,25,20)(13,28,30,17,32,35,31,36)(22,38,29,34,39,33,40,37);(9,13)(10,17)(11,22)(12,28)(14,30)(15,32)(16,33)(18,35)(19,36)(20,37)(21,38)(23,39)(24,29)(25,40)(26,31)(27,34);(11,23)(16,21)(20,27)(22,39)(24,25)(29,40)(33,38)(34,37)

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=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348037);