#6351516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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	#62748	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2804 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,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, 6258712);