#491859

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×D₄	C₂×D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 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:

(1,2,5,8)(3,9,12,15)(4,13,14,7)(6,10,16,11)(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);(1,3,5,12)(2,6,8,16)(4,10,14,11)(7,15,13,9);(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);(3,10)(4,14)(6,15)(7,13)(9,16)(11,12)(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|a^2=g,b^2=g,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^b=de,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399055);