#6351982

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2657 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:

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

Polycyclic presentation:

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

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, 6259178);