#504176

order	512
nilpotency class	4
isoclinism	isoclinic class of #504175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26392
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=ehi,h^d=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411372);