#496887

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265463, #72265476, #72265492, …
maximal normal subgroup of	#72265463, #72265476, #72265492, …
maximal quotient of	#72320485, C₂×#496887, #72343315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404083);