#6359343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359226
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

1540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266539);