#6508685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
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	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18)(19,27)(20,24)(21,33)(22,25)(23,29)(26,36)(28,44)(30,41)(31,42)(32,39)(34,38)(35,40)(37,45)(43,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,38)(26,41)(27,33)(29,36)(31,47)(32,46)(35,45)(39,48)(40,43)(42,44);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,32)(18,24,25,39)(19,28,30,46)(21,35,36,31)(23,37,38,48)(26,42,33,40)(27,43,41,44)(29,47,34,45);(4,12)(7,16)(9,11)(13,15);(3,10)(6,14)(9,11)(13,15)(18,25)(23,38)(24,39)(26,33)(27,41)(37,48)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=h,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6415881);