#6496107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2825 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,27)(20,24)(21,33)(22,25)(23,29)(26,36)(28,44)(30,41)(31,42)(32,39)(34,38)(35,40)(37,45)(43,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,38)(26,41)(27,33)(29,36)(31,47)(32,46)(35,45)(39,48)(40,43)(42,44);(3,9)(6,13)(10,15)(14,16)(17,20,22,32)(18,24,25,39)(19,28,30,46)(21,35,36,31)(23,37,38,48)(26,42,33,40)(27,43,41,44)(29,47,34,45);(3,9)(6,13)(10,15)(14,16)(18,25)(23,38)(24,39)(26,33)(27,41)(37,48)(40,42)(43,44);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)

Polycyclic presentation:

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

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, 6403303);