#6349892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2782 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)(3,11)(4,10)(5,8)(6,9)(7,14)(12,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,41)(33,39)(34,40)(35,46)(37,45)(43,47)(44,48);(1,3,6,12)(2,7,9,16)(4,13,5,14)(8,15,10,11)(17,19,23,32)(18,24,27,38)(20,29,34,44)(21,31,22,30)(25,37,40,47)(26,41,36,28)(33,45,35,43)(39,48,46,42);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,42)(30,43)(31,45)(32,44)(36,46)(38,47)(41,48);(2,9)(3,12)(8,10)(13,14)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,41)(25,39,40,46)(29,43,44,45)(37,42,47,48);(1,4,6,5)(2,10,9,8)(3,13,12,14)(7,11,16,15)(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(42,48)(43,45)

Polycyclic presentation:

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

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, 6257088);