#6351627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351598
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

2664 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,38)(31,39)(32,42)(35,40)(37,41)(43,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,38)(26,29)(27,42)(28,39)(34,47)(35,44)(36,46)(37,45)(40,48)(41,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,41)(27,34)(28,40)(29,43)(31,46)(32,45)(33,44)(39,48)(42,47);(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,29,39,42)(25,34,40,41)(30,44,45,46)(38,47,48,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,32)(24,39)(29,42)(30,45)(31,33)(38,48)(43,47)(44,46)

Polycyclic presentation:

2^9|d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,e^b=eh,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, 6258823);