#6621376

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#471
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528572);