#483294

order	512
nilpotency class	4
isoclinism	isoclinic class of #482975
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62475	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

662 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)(17,18,21,27,20,25,35,43,22,28,37,26,23,29,38,42)(19,30,33,46,39,45,36,41,34,47,32,40,31,44,48,24);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,22,34)(18,24,28,41)(20,31,23,39)(21,36,37,48)(25,40,29,46)(26,44,27,45)(30,42,47,43)(32,35,33,38);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20,22,23)(18,25,28,29)(19,31,34,39)(21,35,37,38)(24,40,41,46)(26,42,27,43)(30,44,47,45)(32,36,33,48);(3,9)(7,14)(10,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390490);