#487514

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
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	#3459	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

721 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,42)(29,41)(31,39)(32,46)(34,38)(35,40)(37,43)(44,48)(45,47);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19,22,29)(18,23,26,38)(20,28,31,44)(21,34,36,30)(24,37,39,47)(25,41,33,27)(32,45,35,43)(40,48,46,42);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20)(18,24)(19,28)(21,35)(22,31)(23,37)(25,40)(26,39)(27,42)(29,44)(30,45)(32,36)(33,46)(34,43)(38,47)(41,48);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(19,29)(20,31)(23,38)(24,39)(27,41)(30,34)(32,35)(40,46)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394710);