#109391

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 16587);