#6351829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	C₄²	C₂⁵
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2758 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259025);