#6496137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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	#62805	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2831 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)(19,29,32,43)(20,33,36,44)(21,28,22,26)(24,37,40,31)(25,38,42,35)(30,39,34,41)(45,46,48,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,37)(22,31)(23,32)(25,41)(26,43)(27,40)(28,29)(30,36)(33,45)(35,46)(38,47)(39,42)(44,48);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,38)(23,36)(24,39)(26,33)(27,42)(28,44)(29,45)(31,46)(32,34)(37,47)(40,41)(43,48);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,37)(20,35,36,38)(24,29,40,43)(25,33,42,44)(30,46,34,47)(39,45,41,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(24,40)(25,42)(26,28)(29,43)(33,44)(39,41)(45,48)

Polycyclic presentation:

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

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, 6403333);