#483562

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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 ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322049, #72322062, #72322123
maximal normal subgroup of	#72322049, #72322062, #72322123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390758);