#6349343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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	#62831	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2615 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,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,d^c=dh,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, 6256539);