#500469

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61284	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,25)(13,27)(14,17)(15,19)(16,23)(18,30)(21,36)(22,33)(24,32)(26,34)(28,39)(29,31)(35,37)(38,40);(2,5)(6,7)(11,21)(12,14)(13,28)(16,31)(17,25)(18,34)(20,36)(22,37)(23,29)(24,38)(26,30)(27,39)(32,40)(33,35);(9,11,15,22)(10,16,19,32)(12,24,14,23)(13,29,30,38)(17,33,25,20)(18,35,27,36)(21,26,37,28)(31,39,40,34);(1,3,4,8)(2,6,5,7)(11,22)(16,32)(20,33)(21,37)(23,24)(29,38)(31,40)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407665);