#483693

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	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 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)(17,18,21,27,22,20,32,35,23,26,36,43,37,34,25,41)(19,24,30,39,31,33,38,47,29,40,45,48,46,28,42,44);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,42)(26,40)(27,39)(28,34)(32,38)(35,47)(36,45)(37,46)(41,44)(43,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20,21,35,22,26,32,43,23,34,36,41,37,18,25,27)(19,28,30,44,31,24,38,39,29,33,45,47,46,40,42,48);(3,9)(7,14)(10,15)(13,16)(18,26)(19,29)(20,34)(27,43)(30,45)(31,46)(35,41)(38,42)

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,c^a=ci,d^a=di,e^a=eg,c^b=ci,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, 390889);