#504170

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411366);