#483664

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 ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 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)(17,18,21,26,22,27,36,42,23,28,37,43,38,44,39,48)(19,24,30,40,31,20,32,34,29,35,45,47,46,33,25,41);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,24)(20,27)(21,30)(22,31)(23,29)(25,39)(26,40)(28,35)(32,36)(33,44)(34,42)(37,45)(38,46)(41,48)(43,47);(17,20,21,34,22,35,36,47,23,33,37,41,38,24,39,40)(18,25,26,19,27,30,42,31,28,32,43,29,44,45,48,46);(2,7)(6,15)(8,12)(9,16)(19,29)(20,33)(24,35)(25,32)(30,45)(31,46)(34,41)(40,47)

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

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, 390860);