#490889

order	512
nilpotency class	4
isoclinism	isoclinic class of #490883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398085);