#474434

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

786 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=egi,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, 381630);