#6351958

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

2716 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)(17,18,24,30)(19,31,35,48)(20,36,32,44)(21,27,40,47)(22,28,23,29)(25,33,43,34)(26,41,42,37)(38,45,39,46);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19,24,35)(18,25,30,43)(20,21,32,40)(22,33,23,34)(26,27,42,47)(28,31,29,48)(36,46,44,45)(37,38,41,39);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20,24,32)(18,26,30,42)(19,21,35,40)(22,37,23,41)(25,27,43,47)(28,44,29,36)(31,45,48,46)(33,38,34,39);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,21)(18,27)(19,32)(20,35)(22,39)(23,38)(24,40)(25,42)(26,43)(28,46)(29,45)(30,47)(31,44)(33,37)(34,41)(36,48);(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,32,41)(21,38,40,39)(25,31,43,48)(26,44,42,36)(27,45,47,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 6259154);