#486768

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
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	C₂²×C₈	C₂⁴
prime radical	#466	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

678 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393964);