#6351522

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

2661 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 6258718);