#483704

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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	#121	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

402 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:

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

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=dgi

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, 390900);