#147253

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54449);