#497844

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405040);