#146290

order	512
nilpotency class	4
isoclinism	isoclinic class of #146290
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10690
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=def,b^2=de,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fgi,g^b=gi,e^c=eg,f^c=fhi,h^c=hi,f^d=fi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53486);