#6349876

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2717 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)(19,28)(20,30)(21,26)(22,27)(23,25)(24,31)(29,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,24)(20,25)(21,30)(22,31)(23,29)(26,28)(27,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,20,23,24)(18,19,25,29)(21,28,22,32)(26,31,27,30);(2,8)(6,12)(7,14)(13,16)(18,25)(19,29)(26,27)(30,31);(3,9)(7,14)(10,15)(13,16)(17,21,23,22)(18,26,25,27)(19,30,29,31)(20,32,24,28)

Polycyclic presentation:

2^9|c^2=h,e^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, 6257072);