#6349208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349137
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	C₄²	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2711 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,f^c=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, 6256404);