#6351619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351598
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	#62825	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2621 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)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,31,41)(20,33,34,42)(21,27,22,26)(24,36,39,32)(25,37,40,35)(30,38,44,48)(43,45,47,46);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,41)(27,29)(28,39)(33,47)(34,44)(35,46)(37,45)(40,48)(42,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,34)(18,25,28,40)(19,30,31,44)(21,35,22,37)(24,38,39,48)(26,33,27,42)(29,43,41,47)(32,45,36,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,34)(25,40)(30,44)(33,42)(35,37)(38,48)(43,47)(45,46);(19,31)(24,39)(29,41)(30,44)(32,36)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

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, 6258815);