#6351800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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

2688 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:

(3,10)(4,5)(7,14)(8,12)(11,13)(15,16)(17,18)(19,29)(20,33)(21,28)(22,27)(23,26)(24,31)(25,35)(30,38)(32,41)(34,42)(36,39)(37,40)(43,46)(44,48)(45,47);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,41)(28,29)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20,23,34)(18,25,26,40)(19,30,32,44)(21,35,22,37)(24,38,39,48)(27,33,28,42)(29,43,41,47)(31,45,36,46);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(20,34)(25,40)(30,44)(33,42)(35,37)(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|c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258996);