#6351693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351670
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

2940 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)(7,13)(12,15)(14,16)(18,23)(20,21)(24,31)(25,26)(28,29)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,30)(29,31);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(19,27)(24,32)(28,29)(30,31);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,11,15,16)(17,20,22,21)(18,25,26,23)(19,28,27,29)(24,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258889);