#491883

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	#33872	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 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)(17,18,21,24)(19,28,30,45)(20,26,23,37)(22,34,33,46)(25,41,39,31)(27,32,40,36)(29,47,38,44)(35,42,48,43);(1,3,5,12)(2,6,8,16)(4,10,14,11)(7,15,13,9)(18,24)(25,39)(26,37)(27,40)(28,45)(34,46)(42,43)(44,47);(17,19,21,30)(18,25,24,39)(20,29,23,38)(22,35,33,48)(26,42,37,43)(27,44,40,47)(28,46,45,34)(31,32,41,36);(3,10)(4,14)(6,15)(7,13)(9,16)(11,12)(17,20,21,23)(18,26,24,37)(19,29,30,38)(22,32,33,36)(25,42,39,43)(27,34,40,46)(28,47,45,44)(31,35,41,48)

Polycyclic presentation:

2^9|a^2=gi,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=dei,e^b=eg,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, 399079);