#489795

order	512
nilpotency class	4
isoclinism	isoclinic class of #489733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396991);