#243919

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

729 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151115);