#243915

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 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:

(1,2,5,8,6,3,4,7)(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);(1,3,6,2)(4,7,5,8)(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);(2,7)(3,8)(4,5)(9,12)(10,17)(11,22)(13,30)(14,28)(15,29)(16,31)(18,36)(19,35)(20,26)(21,34)(23,27)(24,37)(25,38)(32,39)(33,40);(1,4,6,5)(2,7,3,8)(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,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, 151111);