#6349549

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
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
prime radical	#60561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2837 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)(5,10)(6,9)(7,12)(13,16)(14,15)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,16,11)(17,19,22,28)(18,24,26,32)(20,27,21,29)(23,30,25,31);(2,9)(7,16)(8,10)(11,15);(1,4,6,5)(2,10,9,8)(3,13,14,12)(7,15,16,11)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,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, 6256745);