#507379

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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)(9,10,12,14)(11,13,15,16)(17,18,24,32,35,33,36,38,22,20,21,26,23,27,39,34)(19,25,37,46,42,45,44,43,28,31,29,40,30,41,48,47);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,31)(21,29)(22,28)(23,30)(24,37)(26,40)(27,41)(32,46)(33,45)(34,47)(35,42)(36,44)(38,43)(39,48);(11,15)(13,16)(17,20,22,18)(19,25,28,31)(21,34,24,38)(23,27,35,33)(26,36,32,39)(29,43,37,47)(30,45,42,41)(40,48,46,44);(2,6)(5,7)(19,28)(25,31)(29,37)(30,42)(40,46)(41,45)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^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, 414575);