#6352608

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

2974 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:

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

Polycyclic presentation:

2^9|c^2=hi,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, 6259804);