#491074

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
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	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,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=dgi,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398270);