#6496115

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	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2870 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)(17,18,23,27)(19,29,33,39)(20,25,36,43)(21,28,22,26)(24,31,40,32)(30,46,34,45)(35,44,37,42)(38,48,41,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,32)(22,31)(23,33)(25,41)(26,29)(27,40)(28,39)(30,36)(35,47)(37,48)(38,43)(42,46)(44,45);(3,9)(6,13)(10,15)(14,16)(17,20,23,36)(18,25,27,43)(19,30,33,34)(21,35,22,37)(24,38,40,41)(26,42,28,44)(29,45,39,46)(31,47,32,48);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,43,44)(30,47,34,48)(38,46,41,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,40)(25,43)(26,28)(29,39)(38,41)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=hi,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, 6403311);