#497107

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404303);