#126666

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

484 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33862);