#487698

order	512
nilpotency class	4
isoclinism	isoclinic class of #487691
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
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^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,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, 394894);