#6351511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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	C₂×#3559	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2777 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:

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

Polycyclic presentation:

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

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, 6258707);