#507489

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, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=egi,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, 414685);