#243872

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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

820 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)(9,10,15,29,14,19,13,18)(11,20,22,16,24,31,23,32)(12,25,26,35,28,17,27,34)(21,36,39,30,38,40,37,33);(2,6)(3,4)(7,8)(9,11,13,22,14,24,15,23)(10,16,18,31,19,32,29,20)(12,21,27,39,28,38,26,37)(17,33,35,36,25,30,34,40);(2,6)(3,4)(7,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,26)(16,30)(18,35)(19,25)(20,36)(22,39)(23,37)(24,38)(29,34)(31,40)(32,33);(1,3,5,4)(2,6,8,7)(9,13,14,15)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,31,32,20)(17,34,25,35)(21,37,38,39)(30,36,33,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151068);