#497106

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=e,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, 404302);