#6346753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346727
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
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6253949);