#6346771

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	#62820	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,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, 6253967);