#483083

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

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, 390279);