#487794

order	512
nilpotency class	4
isoclinism	isoclinic class of #487659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#534	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

728 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,28)(20,25)(21,34)(22,36)(23,27)(24,30)(26,38)(29,44)(31,43)(32,45)(33,48)(35,42)(37,41)(39,46)(40,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,24)(20,32)(21,29)(22,37)(23,31)(25,40)(26,43)(27,41)(28,36)(30,38)(33,46)(34,39)(35,47)(42,44)(45,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,23,21)(18,25,27,34)(19,29,31,32)(22,35,38,33)(24,39,41,40)(26,42,36,48)(28,44,43,45)(30,47,37,46);(2,7)(6,15)(8,12)(9,16)(17,21,23,20)(18,25,27,34)(19,29,31,32)(22,35,38,33)(24,40,41,39)(26,42,36,48)(28,45,43,44)(30,46,37,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394990);