#479668

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=g,c^2=g,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

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, 386864);