#93746

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₂×C₈	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×D₈	C₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18,19,24,21,25,28,36,22,26,29,37,32,38,41,46,20,23,27,33,30,34,39,43,31,35,40,44,42,45,47,48);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(18,23)(19,27)(25,34)(26,35)(28,39)(29,40)(38,45)(41,47)

Polycyclic presentation:

2^9|a^2=d,c^2=egi,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=cei,d^b=di,e^b=eg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 942);