#6349414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2730 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)(17,18)(19,28)(20,30)(21,25)(22,26)(23,27)(24,31)(29,32);(1,2,6,9)(3,8,11,16)(4,7,5,13)(10,15,12,14)(17,19,23,24)(18,20,27,32)(21,29,22,30)(25,28,26,31);(1,3)(2,8)(4,12)(5,10)(6,11)(7,14)(9,16)(13,15)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,29)(26,30);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(17,21,23,22)(18,25,27,26)(19,29,24,30)(20,28,32,31);(2,9)(7,13)(8,16)(14,15)(17,21,23,22)(18,25,27,26)(19,30,24,29)(20,31,32,28)

Polycyclic presentation:

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

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, 6256610);