#109063

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16259);