#6352638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
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	C₂²×C₄	C₂⁵
prime radical	#62933	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259834);