#6349186

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	#60965	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2772 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,16)(13,15)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,32)(25,36)(30,44)(31,40)(33,39)(35,41)(37,42)(38,45)(43,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,39)(27,40)(28,29)(34,48)(35,47)(36,46)(37,45)(41,44)(42,43);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,20,23,37)(18,25,26,41)(19,30,31,45)(21,36,22,35)(24,38,39,44)(27,34,28,42)(29,43,40,48)(32,47,33,46);(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|c^2=h,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, 6256382);