#507422

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

550 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=i,d^2=e,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, 414618);