#6346786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346762
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	C₂²×C₄	C₂⁵
prime radical	#62823	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2765 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253982);