#243914

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

843 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:

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

Polycyclic presentation:

2^9|a^2=e,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, 151110);