#6349571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
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	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2788 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)(3,11)(4,8)(5,10)(6,9)(7,12)(13,16)(14,15)(17,18,23,28)(19,24,31,39)(20,32,34,42)(21,27,22,26)(25,36,40,33)(29,43,45,47)(30,41,35,38)(37,44,48,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,41)(28,39)(32,47)(33,44)(34,45)(36,46)(40,48)(42,43);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,16,11)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,32)(27,42)(28,40)(30,44)(31,45)(35,46)(38,47)(39,48)(41,43);(2,9)(7,16)(8,10)(11,15)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,45,46)(37,47,48,43);(1,4,6,5)(2,10,9,8)(3,13,14,12)(7,15,16,11)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256767);