#6530954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
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	#62825	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6438150);