#6542402

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	#62927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2947 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)(11,21,25,32)(12,26,29,34)(13,19)(14,20)(15,18)(16,23,31,24)(17,27,33,28)(22,36)(30,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,31)(19,21)(20,32)(26,35)(27,37)(28,39)(29,38)(33,36)(34,40);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,39)(25,38)(31,36)(32,40);(1,3)(2,6)(4,7)(5,8)(10,18)(16,31)(17,33)(19,20)(21,32)(26,34)(30,36)(35,40);(3,7)(6,8)(9,13,15,14)(10,19,18,20)(11,23,25,24)(12,27,29,28)(16,21,31,32)(17,26,33,34)(22,37,38,39)(30,35,36,40)

Polycyclic presentation:

2^9|a^2=fgi,e^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, 6449598);