#483728

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 ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

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,d^a=dg,e^a=eg,d^b=di,e^b=eg,d^c=dgi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390924);