#243909

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
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

778 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:

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

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151105);