#507488

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×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

564 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:

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

Polycyclic presentation:

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

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, 414684);