#170263

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2011	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

831 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77459);