#165831

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

760 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)(14,20)(21,24)(25,26,29,33)(27,30,35,39)(28,31,36,32)(34,37,40,38);(1,2,4,6)(3,8,7,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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,e^c=eh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73027);