#6454821

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	#62858	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3055 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)(11,22)(12,17)(13,18)(14,20)(15,21)(16,24)(19,31)(23,38)(25,33)(26,32)(27,34)(28,35)(29,37)(30,36)(39,40);(9,11,14,24,15,13,31,25)(10,16,19,22,21,33,20,18)(12,26,29,27,28,39,30,23)(17,34,36,32,35,38,37,40);(2,5)(6,8)(9,12,15,28)(10,17,21,35)(11,23,13,27)(14,29,31,30)(16,32,33,40)(18,34,22,38)(19,36,20,37)(24,26,25,39);(9,13)(10,18)(11,15)(12,27)(14,25)(16,19)(17,34)(20,33)(21,22)(23,28)(24,31)(26,30)(29,39)(32,36)(35,38)(37,40);(1,3,4,7)(2,6,5,8)(12,28)(17,35)(23,27)(26,39)(29,30)(32,40)(34,38)(36,37)

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6362017);