#6489626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470049
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	C₂²×C₄	C₂⁵
prime radical	#60561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2924 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(18,23)(19,27)(20,21)(25,26)(29,30)(31,32);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,18,22,25)(19,28,29,24)(20,26,21,23)(27,31,30,32);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,19,22,29)(18,24,25,28)(20,27,21,30)(23,31,26,32);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(18,25)(23,26)(24,28)(31,32);(2,10)(7,16)(8,9)(14,15)(17,20,22,21)(18,26,25,23)(19,27,29,30)(24,32,28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=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, 6396822);