#487844

order	512
nilpotency class	4
isoclinism	isoclinic class of #487715
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

788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,26)(20,29)(21,24)(22,27)(23,28)(25,31)(30,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,21)(18,23,27,32)(20,30,31,28)(24,29,26,25);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,21)(20,31)(23,25)(26,27)(29,32);(2,7)(6,15)(8,12)(9,16)(17,19,22,21)(18,24,27,26)(20,28,31,30)(23,29,32,25)

Polycyclic presentation:

2^9|b^2=g,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,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,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, 395040);