#483526

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,e^b=eg,d^c=di

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321706, #72322049, #72322062
maximal normal subgroup of	#72321706, #72322049, #72322062

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390722);