#148753

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(5,6,7,8)(9,10,12,18)(11,21,23,16)(13,17,27,38)(14,31,28,19)(15,32,29,20)(22,36,24,40)(25,35,30,39)(26,37,33,34);(1,2,4,3)(9,11,12,23)(10,16,18,21)(13,30,27,25)(14,22,28,24)(15,33,29,26)(17,35,38,39)(19,36,31,40)(20,34,32,37);(1,3,4,2)(10,17)(11,22)(13,28)(14,27)(15,29)(16,34)(18,38)(19,20)(21,37)(23,24)(25,33)(26,30)(31,32)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=di,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55949);