#6350663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2791 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,23,27)(19,28,20,30)(21,26,22,32)(24,31,25,29);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10)(17,19,23,20)(18,24,27,25)(21,29,22,31)(26,30,32,28);(17,20,23,19)(18,25,27,24)(21,31,22,29)(26,28,32,30);(2,8)(7,15)(9,13)(10,16)(17,18)(19,24)(20,25)(21,32)(22,26)(23,27)(28,29)(30,31);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,26,27,32)(19,29,20,31)(24,30,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^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,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, 6257859);