#328170

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235366);