#6346712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346696
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

3010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6253908);