#487873

order	512
nilpotency class	4
isoclinism	isoclinic class of #487707
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×C₄⋊C₄	C₂⁴
prime radical	#3492	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

714 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)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,44)(28,38)(29,42)(31,48)(32,43)(35,41)(37,45)(39,46)(40,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,28)(18,23,26,38)(20,31,32,27)(21,29,36,46)(24,40,41,37)(25,42,34,39)(30,44,43,48)(33,47,35,45);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,20,22,32)(18,24,26,41)(19,27,28,31)(21,35,36,33)(23,37,38,40)(25,43,34,30)(29,47,46,45)(39,48,42,44);(2,7)(6,15)(8,12)(9,16)(19,28)(20,32)(23,38)(24,41)(29,46)(30,43)(33,35)(39,42)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,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, 395069);