#6454825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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	#62857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2994 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,17)(13,27)(14,19)(15,20)(16,22)(21,33)(23,31)(24,39)(25,34)(26,36)(28,32)(29,35)(30,38)(37,40);(9,11,13,22,15,23,14,28)(10,16,19,18,20,32,27,31)(12,24,26,21,25,38,29,37)(17,33,35,39,34,40,36,30);(2,5)(6,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,25)(16,30)(18,33)(19,35)(20,34)(22,38)(23,37)(24,28)(27,36)(31,40)(32,39);(9,11)(10,18)(12,21)(13,22)(14,28)(15,23)(16,27)(17,33)(19,32)(20,31)(24,29)(25,37)(26,38)(30,36)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(12,25)(17,34)(21,37)(24,38)(26,29)(30,39)(33,40)(35,36)

Polycyclic presentation:

2^9|b^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,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, 6362021);