#166960

order	512
nilpotency class	3
isoclinism	isoclinic class of #166289
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	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

772 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:

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

Polycyclic presentation:

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

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, 74156);