#491044

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

441 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)(17,18,23,29)(19,28,32,40)(20,34,37,48)(21,30,24,27)(22,25,33,39)(26,41,44,31)(35,45,36,47)(38,42,46,43);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19,24,33,23,32,21,22)(18,25,30,40,29,39,27,28)(20,31,35,46,37,41,36,38)(26,34,42,47,44,48,43,45);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20)(18,26)(19,31)(21,36)(22,38)(23,37)(24,35)(25,34)(27,43)(28,45)(29,44)(30,42)(32,41)(33,46)(39,48)(40,47);(2,8)(3,9)(6,13)(10,15)(17,21,23,24)(18,27,29,30)(19,22,32,33)(20,35,37,36)(25,28,39,40)(26,42,44,43)(31,46,41,38)(34,47,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 398240);