#6352588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352581
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2951 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)(4,14)(8,19)(9,18)(10,15)(12,23)(16,27)(20,25)(21,29)(22,28)(26,31)(30,32);(1,2,5,10)(3,8,11,20)(4,15,14,7)(6,9,17,22)(12,25,23,19)(13,21,24,30)(16,28,27,18)(26,32,31,29);(4,16)(7,18)(12,26)(14,27)(15,28)(19,29)(23,31)(25,32);(1,3,6,13)(2,8,9,21)(4,12,16,26)(5,11,17,24)(7,19,18,29)(10,20,22,30)(14,23,27,31)(15,25,28,32);(3,11)(8,20)(12,23)(13,24)(19,25)(21,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259784);