#6349729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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

2833 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)(4,7)(5,9)(6,10)(8,13)(12,15)(14,16)(17,18,23,28)(19,29,33,38)(20,25,36,42)(21,27,22,26)(24,32,39,31)(30,43,46,47)(34,41,35,40)(37,44,48,45);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,37)(26,29)(27,38)(28,39)(34,44)(35,45)(36,46)(40,47)(41,43)(42,48);(1,3,6,12)(2,8,10,16)(4,13,5,14)(7,15,9,11)(17,20,23,36)(18,25,28,42)(19,30,33,46)(21,35,22,34)(24,37,39,48)(26,41,27,40)(29,43,38,47)(31,45,32,44);(3,12)(8,16)(11,15)(13,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,39,38)(25,40,42,41)(30,44,46,45)(37,47,48,43);(1,4,6,5)(2,9,10,7)(3,13,12,14)(8,11,16,15)(17,21,23,22)(18,26,28,27)(19,32,33,31)(20,35,36,34)(24,38,39,29)(25,41,42,40)(30,44,46,45)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=hi,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,d^c=dh,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, 6256925);