#479294

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ei,c^2=ei,d^2=efi,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, 386490);