#6470035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4208 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,23,30)(12,17,24,32)(13,18,14,26)(16,27,31,22)(21,36,37,35)(25,33,28,34)(29,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,22)(14,27)(15,23)(17,29)(18,30)(19,31)(20,26)(24,37)(25,38)(28,39)(32,40)(33,36)(34,35);(3,7)(5,8)(9,12)(10,17)(11,21)(13,25)(14,28)(15,24)(16,29)(18,33)(19,32)(20,35)(22,38)(23,37)(26,34)(27,39)(30,36)(31,40);(3,7)(5,8)(12,24)(17,32)(21,37)(25,28)(29,40)(33,34)(35,36)(38,39);(2,6)(5,8)(9,13,15,14)(10,18,19,26)(11,22,23,27)(12,25,24,28)(16,30,31,20)(17,33,32,34)(21,38,37,39)(29,36,40,35)

Polycyclic presentation:

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

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, 6377231);