#110470

order	512
nilpotency class	3
isoclinism	isoclinic class of #9781
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

769 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)(9,10,13,18,15,19,14,28)(11,20,23,34,22,16,24,30)(12,17,25,31,27,32,26,38)(21,33,36,40,35,29,37,39);(2,6,8,7)(9,11,14,24,15,22,13,23)(10,16,18,30,19,20,28,34)(12,21,26,37,27,35,25,36)(17,29,31,39,32,33,38,40);(2,6,8,7)(9,12,13,25,15,27,14,26)(10,17,18,31,19,32,28,38)(11,21,23,36,22,35,24,37)(16,29,30,39,20,33,34,40)

Polycyclic presentation:

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

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, 17666);