#449461

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = 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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=gi,d^2=egi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=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, 356657);