#475999

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

816 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,21,23,24)(25,26,29,32)(27,31,30,38)(28,35,37,40)(33,34,39,36);(2,5)(6,8)(17,19,20,23)(18,21,22,24)(25,27,29,30)(26,31,32,38)(28,36,37,34)(33,40,39,35);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(19,23)(21,24)(26,32)(31,38)(33,39)(35,40);(3,7)(6,8)(25,28,27,34,29,37,30,36)(26,33,38,40,32,39,31,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,d^c=de

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, 383195);