#6351859

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2770 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)(3,13)(4,16)(5,11)(6,10)(7,12)(8,15)(9,14)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,44)(31,38)(32,40)(33,42)(34,41)(36,39)(37,45)(43,46)(47,48);(1,3,7,4)(2,8,12,9)(5,14,6,15)(10,16,11,13)(17,19,23,31)(18,24,27,39)(20,29,34,46)(21,30,22,36)(25,37,41,48)(26,38,35,28)(32,45,33,47)(40,43,42,44);(1,4,7,3)(2,9,12,8)(5,14,6,15)(10,16,11,13)(17,20,23,34)(18,25,27,41)(19,29,31,46)(21,33,22,32)(24,37,39,48)(26,42,35,40)(28,43,38,44)(30,47,36,45);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,16,9,13)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,38,39,28)(25,40,41,42)(29,45,46,47)(37,43,48,44);(1,5,7,6)(2,11,12,10)(3,15,4,14)(8,16,9,13)(19,31)(24,39)(28,38)(29,46)(30,36)(37,48)(43,44)(45,47)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,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, 6259055);