#6630851

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538047);