#475720

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=egh,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382916);