#243878

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 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)(9,10,14,18,15,19,13,27)(11,20,22,32,23,31,29,16)(12,24,25,34,26,17,28,35)(21,36,38,33,39,40,37,30);(2,6)(3,4)(7,8)(9,11,13,22,15,23,14,29)(10,16,27,20,19,32,18,31)(12,21,28,38,26,39,25,37)(17,33,34,40,24,30,35,36);(2,7)(3,4)(6,8)(9,12)(10,17)(11,21)(13,28)(14,25)(15,26)(16,30)(18,35)(19,24)(20,36)(22,38)(23,39)(27,34)(29,37)(31,40)(32,33);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,27)(11,22,23,29)(12,25,26,28)(16,31,32,20)(17,34,24,35)(21,37,39,38)(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,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

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, 151074);