#6349296

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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

2861 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(18,22)(19,25)(20,27)(24,28)(26,29)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,19,21,26)(18,23,24,31)(20,29,27,25)(22,30,28,32);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15);(1,4,6,5)(2,8,10,9)(3,13,14,12)(7,11,16,15)(18,24)(22,28)(23,31)(30,32)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=hi,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,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, 6256492);