#104993

order	512
nilpotency class	3
isoclinism	isoclinic class of #104973
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12189);