#6452940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437445
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	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3142 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)(4,7)(5,8)(9,10)(11,22)(12,18)(13,19)(14,20)(15,30)(16,21)(17,26)(23,32)(24,34)(25,33)(27,38)(28,35)(29,36)(31,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,23)(13,24)(14,26)(15,25)(16,27)(18,34)(19,37)(20,38)(21,33)(22,30)(28,40)(29,39)(31,36)(32,35);(1,3,5,4)(2,7,8,6)(9,12,16,29)(10,18,21,36)(11,23,27,39)(13,14,28,15)(17,31,33,34)(19,20,35,30)(22,32,38,37)(24,26,40,25);(1,4,5,3)(2,7,8,6)(9,13)(10,19)(11,24)(12,14)(15,29)(16,28)(17,32)(18,20)(21,35)(22,34)(23,26)(25,39)(27,40)(30,36)(31,38)(33,37);(9,14,16,15)(10,20,21,30)(11,25,27,26)(12,28,29,13)(17,22,33,38)(18,35,36,19)(23,24,39,40)(31,32,34,37)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,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, 6360136);