#6349768

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349761
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

2973 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,26)(21,24)(22,25)(23,29)(28,32)(30,31);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,22,30)(18,23,25,32)(20,28,21,29)(24,31,26,27);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(18,25)(23,32)(24,26)(27,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,21)(18,26,25,24)(19,28,30,29)(23,27,32,31)

Polycyclic presentation:

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

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, 6256964);