#6351554

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	#62752	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2623 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)(17,18,23,27)(19,29,32,39)(20,33,35,41)(21,26,22,28)(24,36,40,31)(25,37,42,34)(30,38,46,48)(43,44,47,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,23,32)(18,24,27,40)(20,30,35,46)(21,36,22,31)(25,38,42,48)(26,29,28,39)(33,47,41,43)(34,45,37,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,33)(29,43)(31,44)(32,46)(36,45)(39,47)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,40,29)(25,41,42,33)(30,44,46,45)(38,47,48,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,27)(24,40)(25,42)(26,28)(29,39)(33,41)(38,48)(43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,e^2=i,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, 6258750);