#166184

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#52280	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

903 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,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, 73380);