#243844

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

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:

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

Polycyclic presentation:

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

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, 151040);