#6351789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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

2961 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)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3,6,13)(2,7,9,15)(4,12,5,14)(8,11,10,16)(20,29)(24,28)(25,32)(27,30);(4,5)(8,10)(11,16)(12,14)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(18,23)(22,31)(24,28)(25,32);(2,9)(7,15)(8,10)(11,16)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280141
maximal normal subgroup of	#72280141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258985);