#6684559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61222	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591755);