#6351474

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2616 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,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, 6258670);