#105084

order	512
nilpotency class	3
isoclinism	isoclinic class of #105073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,e^c=egh,f^c=fgi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12280);