#6470065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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

4206 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)(9,10,15,20)(11,21,23,34)(12,26,28,32)(13,18,14,19)(16,24,31,25)(17,29,33,27)(22,30,36,40)(35,38,39,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,30)(18,21)(19,34)(20,31)(26,39)(27,37)(28,36)(29,38)(32,35)(33,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,32)(19,26)(20,33)(21,35)(23,36)(24,37)(25,38)(31,40)(34,39);(3,7)(5,8)(11,23)(16,31)(21,34)(22,36)(24,25)(30,40)(35,39)(37,38);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,27,28,29)(16,21,31,34)(17,32,33,26)(22,37,36,38)(30,35,40,39)

Polycyclic presentation:

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

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, 6377261);