#6359655

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#61393	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1574 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266851);