#507679

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

585 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,28,20,33,35,40,21,27,36,47,23,26,38,46)(19,29,32,44,34,43,48,45,31,24,39,42,30,41,37,25);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,30)(25,40)(26,43)(27,29)(28,42)(33,41)(35,48)(36,39)(37,38)(44,47)(45,46);(9,11,12,16)(10,14,13,15)(17,19)(18,25)(20,30)(21,31)(22,37)(23,34)(24,40)(26,44)(27,45)(28,41)(29,46)(32,38)(33,42)(35,39)(36,48)(43,47);(2,6)(5,7)(10,13)(14,15)(17,20,21,23)(18,26,27,33)(19,30,31,34)(22,35,36,38)(24,41,29,43)(25,44,45,42)(28,46,47,40)(32,37,39,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=g,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414875);