#6349178

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	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2754 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)(17,18,23,26)(19,29,33,40)(20,34,37,42)(21,27,22,28)(24,32,39,31)(25,36,41,35)(30,44,47,38)(43,46,48,45);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(17,19,23,33)(18,24,26,39)(20,30,37,47)(21,32,22,31)(25,38,41,44)(27,29,28,40)(34,48,42,43)(35,46,36,45);(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,42)(28,34)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(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)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,35,37,36)(24,40,39,29)(25,42,41,34)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=h,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, 6256374);