#6351866

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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	#62824	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2810 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259062);