#6349678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349612
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

2851 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18)(19,29)(20,34)(21,28)(22,26)(23,27)(24,33)(25,36)(30,44)(31,40)(32,39)(35,42)(37,41)(38,46)(43,45)(47,48);(2,7)(4,5)(8,14)(9,10)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,38)(26,39)(27,40)(28,29)(34,48)(35,45)(36,47)(37,46)(41,43)(42,44);(1,3,6,11)(2,8,10,15)(4,12,5,13)(7,14,9,16)(17,20,23,37)(18,25,27,42)(19,30,32,46)(21,36,22,35)(24,38,40,44)(26,34,28,41)(29,43,39,48)(31,47,33,45);(3,11)(8,15)(12,13)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,33)(20,35,37,36)(24,39,40,29)(25,41,42,34)(30,45,46,47)(38,43,44,48);(1,4,6,5)(2,7,10,9)(3,12,11,13)(8,14,15,16)(18,27)(19,32)(25,42)(26,28)(30,46)(31,33)(34,41)(45,47)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6256874);