#6349707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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

2727 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:

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256903);