#487504

order	512
nilpotency class	4
isoclinism	isoclinic class of #486891
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	#3460	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

746 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,26)(20,30)(21,25)(22,32)(23,28)(24,29)(27,31);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19,23,22)(18,24,28,30)(20,31,29,21)(25,26,27,32);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20,23,29)(18,25,28,27)(19,21,22,31)(24,32,30,26);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,19,23,22)(18,26,28,32)(20,21,29,31)(24,27,30,25)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394700);