#6349862

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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	#62830	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2855 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,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, 6257058);