#6351802

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
prime radical	#62755	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2689 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)(17,18,23,28)(19,29,33,39)(20,34,35,42)(21,27,22,26)(24,31,40,32)(25,37,41,36)(30,38,44,48)(43,46,47,45);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,39)(27,29)(28,40)(34,47)(35,44)(36,45)(37,46)(41,48)(42,43);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,42)(28,41)(29,43)(31,45)(32,46)(33,44)(39,47)(40,48);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(20,35)(25,41)(30,44)(34,42)(36,37)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,35,37)(24,39,40,29)(25,34,41,42)(30,45,44,46)(38,47,48,43)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,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, 6258998);