#6351063

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258259);