#496885

order	512
nilpotency class	4
isoclinism	isoclinic class of #483919
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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(9,10,16,19,14,20,13,21)(11,22,15,30,25,17,24,33)(12,26,28,32,29,34,31,38)(18,27,37,23,36,39,35,40);(2,5)(3,7)(9,11,16,15,14,25,13,24)(10,17,19,33,20,22,21,30)(12,27,28,23,29,39,31,40)(18,34,37,38,36,26,35,32);(1,3,4,7)(2,6,5,8)(9,12,14,29)(10,18,20,36)(11,23,25,40)(13,28,16,31)(15,27,24,39)(17,32,22,38)(19,35,21,37)(26,30,34,33);(2,5)(6,8)(9,13,14,16)(10,19,20,21)(11,24,25,15)(12,28,29,31)(17,33,22,30)(18,35,36,37)(23,39,40,27)(26,38,34,32)

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,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	#72265451, #72265476, #72265504, …
maximal normal subgroup of	#72265451, #72265476, #72265504, …
maximal quotient of	#72343340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404081);