#6349365

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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2749 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:

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

Polycyclic presentation:

2^9|a^2=h,c^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,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, 6256561);