#6445799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi

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, 6352995);