#6542405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#62944	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2974 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449601);