#6365489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365322
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272685);