#6358559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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

1543 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,28)(19,29,32,38)(20,25,22,27)(21,26,35,44)(24,36,40,30)(31,45,47,42)(33,41,37,43)(34,48,46,39);(17,19,23,32)(18,24,28,40)(20,30,22,36)(21,34,35,46)(25,38,27,29)(26,42,44,45)(31,37,47,33)(39,41,48,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,22)(18,25,28,27)(19,30,32,36)(21,33,35,37)(24,38,40,29)(26,41,44,43)(31,46,47,34)(39,42,48,45);(17,21)(18,26)(19,31)(20,33)(22,37)(23,35)(24,39)(25,41)(27,43)(28,44)(29,45)(30,46)(32,47)(34,36)(38,42)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,22)(18,27,28,25)(19,30,32,36)(21,33,35,37)(24,29,40,38)(26,43,44,41)(31,46,47,34)(39,45,48,42)

Polycyclic presentation:

2^9|a^2=h,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,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265755);