#167820

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

754 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,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, 75016);